I'm not familiar with the specifics of 4k sources or streaming since I don't do it, but I believe you could do this the following way:
1) Set your canvas and output resolution to 1080p
2) Create your first scene, call it "Scene A" or something similar.
3) Add your 4k video source to Scene A. Position and scale it, if needed, so that the first portion of this source is visible and framed how you want it.
4) Now add a second scene. Call it "Scene B" or something appropriate.
5) Add a video source to this scene. Choose an existing source from the list, and then position that so that the a new portion of of your 4K sized source is visible within this scene.
Repeat steps 4-5 until you have as many scenes as you need.
To do your "autopilot" install this:
https://obsproject.com/forum/resources/advanced-scene-switcher.395/
You should be able to configure this plugin to cycle between your scenes on a simple timer if you wish.