Graceful shutdown is handled as part of the existing S6 termination process, using a /etc/cont-finish.d script.
Nginx will attempt to drain active workers, while rejecting new connections. The drain timeout is controlled by S6_KILL_FINISH_MAXTIME, which corresponds to the length of time the supervisor will wait for the script to run during shutdown. This value defaults to 55s, which deliberately less than an downstream load balancers default max connection length (60s). Each upstream's timeout must be less than the downstream, for sanity and lack of timing precision.