My followers no longer follow me around the map. When I switch to another character, my main character follows them normally, but the others continue to stand around wherever they were before. Is this a common problem? What can I do to fix it?

This has happened to me several times. I don't know what's causing it.

Try leaving that area and then entering it again (that usually works for me). Reloading a slightly earlier save may help too.

A few times I was able to jog them back to life (and following) by playing with the Party AI button (turning it off, than on) and switching characters.

Usually, I leave the area, then return.
For me this happened in the first dungeon (the swamp temple) and in the outdoors area on the way to the Ironfist stronghold. It doesn't happen often, as far as I know.

EDIT: Broadcast command: "Follow me" might work too.

Party AI on and Broadcast Command: Follow Me usually works, as Loreseeker says.

However, Some Cutscenes are known to set your companions in Puppet Mode.

You might want to check their Behaviour tab on their Character sheet.

also, in patches 1.10 and later, companions don't follow automatically. you have to select them first (like in baldur's gate etc). i recommend you make a hotkey "select all" in options menu, and the problem is solved.