The issues you are concerned with involve two interelated issues.
The ability of the wire to transmit the current with a feasible voltage drop, and the ability
of the wire to transmit the current without overheating.
Assuming they are copper wires, it is easy to find the resistance per metre of the wire.
If you assume a current of 200 mA ( or any other figure you care to use ), you can figure
out the voltage drop in the wire over the length you are going to use. You can also figure
out the voltage drop. If the motor needs, say, 12 V, and you put 12 volts in at the supply end
but only get 9 volts at the motor end because of the losses in the wires, the motor may struggle
and draw even more current.
Once you know the resistance of the wires, you can also consider how much they will heat up
under sustained operation. Worst case would be almost continuous operation but with frequent
short interruptions, so you get the inrush current again right after each interruption. You can
then figure out the heat loss through the wire insulation, and consider the ventilation, to determine
the temperature the wire might reach.
Not simple, but not rocket science either.
There is no need to try to reconnect new wires inside the device. There is nothing stopping you
connecting thicker wires to the existing leads, just outside the device.