There are a couple of sites that mention this kind-of sort-of trick to kick ARD in the head, but I thought I'd make a note here for my own quick reference in one place. These notes should work on El Capitan (10.11) and Sierra (10.12).
Summary: Remove cached settings from ARD, remove network DNS/ARP caches on machine, kick ARD in the head...
- In ARD, go to the All Computers list, highlight the machine names and delete them.
- Quit ARD.
- Flush DNS cache: sudo dscacheutil -flushcache;sudo killall -HUP mDNSResponder
- Flush ARP: sudo arp -ad
- Kick ARD in the head by restarting the ARD agent (on clients): /System/Library/CoreServices/RemoteManagement/ARDAgent.app/Contents/Resources/kickstart -restart -agent
Start Remote Desktop again and re-scan the network. Because the clients were removed, attempting to view/connect may require you to re-enter credentials.
Also keep in mind I noted testing this on El Capitan and Sierra. Another annoyance with OS X releases is that the syntax/procedure for flushing DNS changes alarmingly often, so it may take some Googling if your release is different.
The last note I have is that if this doesn't work, check that a network hiccup didn't force the client's wireless to shrug its shoulders and give up, meaning that the actual problem all along was that the client was not able to be remotely managed over the network all along.
Whoops.
