Pairing battery voltages with motor voltages

I have a 48v scooter with a weak 10amp controller and a 72v scooter with a 20a controller. I want to try using the new hobby king graphene lipo packs with it, but how do you determine which battery you need? Either than the physical size and the Ah ratings, which I get. Is the voltage listed on the battery the minimum voltage the battery can go to? Same with the 48v motor?
I'm just used to voltages for cell phones, laptops and such where you need to be using the set voltage it's made for. But it seems with hub motors it's a range, and that's what I don't know. Also didn't know till I started trying to figure this out that batteries' voltages changed over time. I always thought the amount of amperage it could put out sagged over time but the voltage stayed the same.

Anyways thanks again for your answers, slowly figuring out how to switch away from the factory SLA batteries.

You probably want to post this in the Electric Scooter section. I'm sure someone here could explain this, but they have DONE it and you will have examples to look at.

https://endless-sphere.com/forums/viewtopic.php?f=10&t=88321

please do not double post. thx.