> Попробуйте включить debug log в nginx'е, возможно происходящее
станет понятнее.

Понятнее не стало.

-------------------------------------------------
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 using configuration "/"
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http cl:-1 max:104857600
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 rewrite phase: 3
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 post rewrite phase: 4
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 generic phase: 5
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 generic phase: 6
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 generic phase: 7
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 access phase: 8
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 access phase: 9
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 access phase: 10
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 post access phase: 11
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 generic phase: 12
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 try files handler
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 trying to use file: "@backend" "/var/www/site/www@backend"
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 test location: "@backend"
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 using location: @backend "/missing/?"
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 rewrite phase: 3
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http script var
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http script value: "_ind.html"
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http script set $handler
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http script complex value
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http script copy: "_action="
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http script var: "/missing/"
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http script copy: "&"
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http script set $querystring
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 post rewrite phase: 4
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 generic phase: 5
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 generic phase: 6
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 generic phase: 7
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 access phase: 8
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 access phase: 9
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 access phase: 10
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 post access phase: 11
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 generic phase: 12
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 generic phase: 13
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http init upstream, client timer: 0
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 epoll add event: fd:51 op:3 ev:80002005
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http map started
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http script var: "http"
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http script copy: "|"
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http script var: "GET"
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http script copy: "|"
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http script copy: "|"
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http script copy: "|"
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http script copy: "|"
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http script var: "/missing/"
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http cache key: "http|GET||||/missing/"
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http map started
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http script var: "/missing/"
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http map started
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http script copy: "|"
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http script copy: "|"
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http map: "||" ""
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http script var: ""
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http map: "/missing/" ""
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http script var: ""
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 add cleanup: 00005594C08CB230
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http file cache exists: 0 e:1
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 cache file: "/var/www/site/cache/pages/49/1d/2756424aaf76d1f24c982e863d611d49"
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 add cleanup: 00005594C08CB288
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http file cache fd: 52
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 read: 52, 00005594C08CB308, 475, 0
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http upstream cache: 0
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 posix_memalign: 00005594C08CB660:4096 @16
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http fastcgi record byte: 01
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http fastcgi record byte: 06
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http fastcgi record byte: 00
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http fastcgi record byte: 01
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http fastcgi record byte: 00
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http fastcgi record byte: 4A
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http fastcgi record byte: 06
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http fastcgi record byte: 00
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http fastcgi record length: 74
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http fastcgi parser: 0
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http fastcgi header: "Content-Type: text/html; charset=UTF-8"
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http fastcgi parser: 0
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http fastcgi header: "Status: 200"
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http fastcgi parser: 0
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http fastcgi header: "Content-Length: 0"
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http fastcgi parser: 1
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http fastcgi header done
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http file cache send: /var/www/site/cache/pages/49/1d/2756424aaf76d1f24c982e863d611d49
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 HTTP/1.1 200
Server: nginx
Date: Mon, 09 Apr 2018 15:21:23 GMT
Content-Type: text/html; charset=UTF-8
Content-Length: 0
Connection: keep-alive
Expires: Mon, 09 Apr 2018 16:21:23 GMT
Cache-Control: max-age=3600
X-XSS-Protection: 1; mode=block
X-Frame-Options: DENY
X-Content-Type-Options: nosniff
Strict-Transport-Security: max-age=31536000

2018/04/09 18:21:23 [debug] 29576#29576: *3395329 write new buf t:1 f:0 00005594C08CB8E0, pos 00005594C08CB8E0, size: 354 file: 0, size: 0
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http write filter: l:0 f:0 s:354
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http output filter "/missing/?"
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http copy filter: "/missing/?"
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http postpone filter "/missing/?" 00007FFF2E0EA068
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 write old buf t:1 f:0 00005594C08CB8E0, pos 00005594C08CB8E0, size: 354 file: 0, size: 0
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 write new buf t:0 f:0 0000000000000000, pos 0000000000000000, size: 0 file: 475, size: 0
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http write filter: l:1 f:0 s:354
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http write filter limit 0
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 writev: 354 of 354
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http write filter 0000000000000000
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http copy filter: 0 "/missing/?"
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http finalize request: 0, "/missing/?" a:1, c:3
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http request count:3 blk:0
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http finalize request: -4, "/missing/?" a:1, c:2
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http request count:2 blk:0
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http finalize request: -4, "/missing/?" a:1, c:1
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 set http keepalive handler
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http close request
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http log handler
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 run cleanup: 00005594C08CB288
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 file cleanup: fd:52
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 run cleanup: 00005594C08CB230
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http file cache cleanup
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 http file cache free, fd: 52
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 free: 00005594BE9D5CA0, unused: 0
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 free: 00005594BE9D6CB0, unused: 3
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 free: 00005594C08CA650, unused: 0
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 free: 00005594C08CB660, unused: 2715
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 free: 00005594BE9D5890
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 hc free: 0000000000000000
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 hc busy: 0000000000000000 0
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 tcp_nodelay
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 reusable connection: 1
2018/04/09 18:21:23 [debug] 29576#29576: *3395329 event timer add: 51: 65000:1523287348578
-------------------------------------------------

> Кроме того, имеет смысл показать полную минимальную конфигурацию, с которой воспроизводится проблема.

Попробую, конечно.
Пока надеюсь выяснить причину на боевой конфигурации.