Although Multiplayer Respawn Points cannot themselves be attached to a mover, the same effect can be achieved using Pure Faction's teleport trigger functionality. This uses the same method we would use to create teleporters.

Prerequisite knowledge: How to create basic movers using keyframes, how to make a keyframe trick

1) Create a room outside of your map for your spawn. I would recommend putting each spawn point in its own individual room. I normally use a 4x4x4 black box-shaped air brush for this.
2) Surround this spawn point with a Trigger Door object with "Trigger resets after" set to 0.01s. Be sure to set the "Is Teleport Trigger" option found in the bottom right hand corner of the Trigger Properties window. If you do not see this option, please use REDPF.
3) Create a keyframe trick somewhere in your map, and link the Trigger Door we just created to the first (gold) keyframe of this trick.
4) Create a "Teleport_Player" event from "Events-> Modifiers" in Object mode, and note the UID of this event.
NOTE: An object's UID can be determined by looking in the REDPF console (bottom left hand corner of the editor) when the object is selected.
5) Put the event's UID in the "Trigger Event With UID" input box in the properties of the second keyframe in the keyframe trick. Now when the player spawns, he will be automatically teleported to the location of your "Teleport_Player" event.
6) Create a new mover containing the "Teleport_Player" event. Players will now "spawn" wherever you have moved the event with this mover.

A few examples of where this can be used:
-Run maps where players automatically respawn at each checkpoint once they've been unlocked
-"Assault-style" CTF maps, where player respawns are automatically moved up when an objective has been completed
-Pseudo-random spawn points