Can small voltage difference cause large current flow?

This is tough to describe but please help

In a design that I have there are 3 positive leads which power 2 seperate boards. (also 3 negative and ground but we can ignore them for now)

I have the current split from 1 of the 3 leads to power both boards, and the other 2 leads just power one board. So:
Lead 1: Powers board 1
Lead 2: Powers board 1 and 2
Lead 3: Powers board 2

Lead 1 is shorted to lead 2 through board 1, and same goes for lead 2 and 3 through board 2.

Sometimes only 1 board is on, and i would like the current to be shared by all 3 leads. Someone (who knows more than me) is trying to convince me that all of the current will flow through 2 of the leads when only 1 board is on, because the voltage drop along the 1 wire through the board and back up to the 2nd lead (maybe 0.1V) will render that wire useless.

Does that make sense? the idea is that all the wires would carry enough current to have a the same voltage, right?

The current will divide between the wires in parallel proportional to the relative wire resistance back to the supply, thus all wires will carry current regardless of which boards are powered. But obviously the percentage that each carries depends upon the board that's powered since the relative resistance of the ground current back to the supply varies with the actual board source location of the ground current and the length of the three paths back to the supply. Does that make sense?

Your "someone" may (or may not) know more than you, but he doesn't understand resistance and parallel current flow.