For clarification the battery provides voltage and the ESC draws Amps.
Meaning there is a valid input voltage range, something like 2S-4S for every ESC. Using any voltage outside that range is not characterized and we don't know what will happen. It might work fine or it might burst into flames, I dunno, but it's not covered by any warranty and you do it at your own risk.
The ESC draws current and will use what is available as needed. Think of this as drinking from a straw. The glass you are drinking from doesn't push the liquid into the straw. You can suck a little bit or you can suck a lot. At some point you reach max suction for the straw/liquid setup and you don't get any more.
This is a very brief description and there area bunch more details in this.