I have just tried to deploying multiple machine ArcGIS Server site on Docker last month. Here are my deployment steps:
(1) I use vmware created three VMs。One is manager host, the others are node hosts.
(2) I installed docker swarm and created a cluster between these VMs. The doc how install swarm is here:
Create a swarm for development
(3) I used a bridge network to accomplish communicate with each docker containers on multiple hosts. The network between docker containers which can across multiple hosts is very important. There are many third software,for example: wave,pipework,open vswitch etc. This is my network architecture diagram.
(4) Create ArcGIS Server base image. I used the official centos 7 image,and installed ArcGIS Server manually.
(5) Deploying multiple machine site by docker.
All of these steps,I think the network is the most important.The ArcGIS Server containers must can communicate with each other directly, not use port mapped from host. Like in the example above, container A(IP:192.168.10.2) can ping other containers(IP:192.168.11.2 and IP:192.168.12.2).
Please forgive me for my bad english.
I hope this can help you.