... who then send me the specific geometries...
Usually, you need to talk to your PCB manufacturer and sometimes it may take weeks to agree on a stackup.
... what happens if I need to change manufacturers? ...Is there a way to make things generic enough? ...
From my experience, if you get a stackup from a Chinese company, most of the other companies can make the PCB. Ask for standard materials. But you are right. There are companies what give you stackup from non-standard materials and then you need to re-define the stackup or you need to stick with them. So be careful.
Here are some ideas for the stackups which we use and can be manufactured by different PCB manufacturers:
Download PCB Stackups – Free for your Projects >If what new manufacturer uses slightly different materials and processes, do I then need to change the Design Rules in my CAD Software to accommodate the new geometries?
If your stackup is good, many manufactures can manufacture it with no changes - that should be the goal. If we notice, that there is a problem with manufacturing a PCB, we always update our documentation (you always want to be sure, that batches of your boards are always manufactured the exactly same way, otherwise you will not be able to guarantee quality - if every manufacturer adjust the PCB their way, your boards will be different between batches and may behave differently)