Looking for recommendations on semi-remote KVM switches

I need to spec out a few KVM/remote systems for control of a few computer systems. This would be used in a broadcast control room for a medium sized church where the computers needed to operate the system would be housed in a server room just a few feet below the broadcast control room.

Must have:
-Simple connectivity. Cat 6a or fiber
-USB 3
-Multiple USB ports, or support for a hub
-DVI/HDMI/DP with minimum 1920x1080 resolution
-zero latency

-Multiple computers connected to multiple displays, almost in a matrix format. They don’t need to have mirrored connections, but it would be nice if that were possible.
-4k resolution
-support for more than 1 monitor

I don’t think these are unreasonable, but last time I set up a KVM it was VGA/PS2 based. I also don’t know trusted brands, and I don’t know what to look for as far as “gotchas” and features.