Assuming all 3 clips have the same properties, basic syntax is
ffplay -f lavfi -i movie=filename="/dev/video0":f=v4l2,setpts=PTS-STARTPTS[v1];movie=filename="/dev/video1":f=v4l2,setpts=PTS-STARTPTS[v2];movie=filename="/dev/video2":f=v4l2,setpts=PTS-STARTPTS[v3];[v2][v3]overlay=enable='between(mod(t,15),10,15)'[v23];[v1][v23]overlay=enable='between(mod(t,15),5,15)'
This will show 5 seconds of video0 followed by 5s of video1 followed by 5s of video2.
Edit by OP: Using this suggestion, I was able to get this working for 4 cameras.
ffplay -f lavfi "movie=/dev/video0:f=video4linux2, setpts=PTS-STARTPTS [zero];movie=/dev/video1:f=video4linux2, setpts=PTS-STARTPTS [one];movie=/dev/video2:f=video4linux2, setpts=PTS-STARTPTS [two];movie=/dev/video3:f=video4linux2, setpts=PTS-STARTPTS [three];[one][zero]overlay=enable='between(mod(t,20),5,10)'[conn1];[conn1][two]overlay=enable='between(mod(t,20),10,15)'[conn2];[conn2][three]overlay=enable='between(mod(t,20),15,20)'[out]"
It defines 4 camera inputs, then does 3 video segments. The first video segment takes cams 0 and 1 and overlays 1 on top of 0 (but only for seconds 5-10). The second segment overlays cam 2 on the stream of cams 0+1 (but only for seconds 10-15). The third segment overlays cam 3 on the combined stream of cams 0+1+2 (but only for seconds 15-20). Each stream is only shown during its "turn" in the loop (again, multiples of 0-5, 5-10, 10-15, 15-20), governed by the overlay commands and the between+modulus magic suggested by Gyan.