[...] can breakers keep a consistent amp cutoff across different voltages?
In North America, many places have 120 V/240 V service and there are "slash rated" 120/240 V circuit breakers which can be used with such installations. The voltage before the slash is the maximum voltage to ground in the system (the 240 V power company transformer secondary is center-tapped and the tap is grounded thus the max volts to ground is 120). You can also get "straight" rated 240 V breakers for use in 'true' 240 V to ground circuits. These will be of more robust construction with insulation rated for the higher voltage. You could use one of these in a 120 V circuit if you wanted, but it would be a needless expense. For either type of breaker, the rated current marking in amps is the same for either a 120 V or 240 V circuit.