How can I get a dynamic ground plane?

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I want it to move with a constant linear or angular velocity

This tutorial goes over various ways of moving objects in simulation with a given velocity:

https://gazebosim.org/tutorials?tut=s...

Note that if your plane has a collision shape, it will carry the models laying on top of it. You can move just a model that has no collision but has a visual in case you just wan the visual to move.

Another possible way of doing this is with scripted trajectories:

gazebosim.org/tutorials?tut=set_velocity

Is it also possible to have a video as ground plane?

Yes, here's some explanation on how to do it with ROS 1 / 2:

https://github.com/ros-simulation/gaz...

You could also use a texture that changes over time, sort of like a GIF, here's an example:

https://github.com/chapulina/simslide...