A connector between shapes?

Yala,

If compatible with your workflow, there is a simple way which would be quite easy to use for straight connectors, whereas curved connectors would/could need adjustment of shape (Handles), and it can also be used for multiple connectors starting/ending in the same shape(s), the key being that each connector is actually unconnected to the shapes that it connects.

1) Create each connector as an open path starting/ending exactly on (the boundaries of) the shapes that it is to visually connect; Smart Guides are your friends saying path when you are within snapping distance (or anchor if you wish to snap it to Anchor Points); maybe you actually have done that already; you can ClickDrag with the Line Segment Tool or Click with the Pen Tool Pen Tool (remember to create a new path each time, you can press P), or ClickDrag with the Pen tool for curved ones;

2) For each shape you wish to move, first deselect everything (you can click an empty spot somewhere), then with the Direct Selection Tool simply ClickDrag over the shape including the start/end Anchor Point of the connector(s) on the shape, then let go of the mouse/whatever to select, then ClickDrag the selection to the desired position.

This will move the shape along with (all) the start/end point(s) of the connector(s), (each of) which will retain its position on (the boundary of) the shape, keeping the connexion(s).

That would need to be hacked and then it might not be very stable. Such as this: https://youtu.be/Pr4_k6y9TNY