How to move components with tracks connected?

Don’t you mean burgers, er, gerbers? :crazy_face: Ok, OP has solved the problem. Time to move on.

Those are also certified to be good because they do not contain an “F”.

And now I can reproduce both behaviors at the same time
kicad-ezgif.com

Switch the footprint pads to outline mode. There may be traces not connected directly to the centre of the pad. The drc check will not flag this problem.

Yes, the trace that does not move with the part is not connected directly to the center of the pad, it just randomly cuts through it.

How do you switch the footprint pads to outline mode? How does that help indicate a connection is present?

One of switches on the left. Other will switch tracks into outline mode.

You see if center of last track segment ending arc is in center of pad. May be you are not 100% sure but mostly sure.