Let's say three people are in a conflict:

Alex is rolling d12/d10
Bert is rolling d10/d8
Charlie is rolling d8/d6

Charlie lucks out and gets 7/5 (7 on the d8, 5 on the d6)
Bert rolls 6/4 (6 on the d10, 4 on the d8)
Alex rolls 6/4 (6 on the d10, 4 on the d12)

Now, Bert and Alex are going to re-roll their lower result to see which goes first.

Bert rolls 5 on the d8
Alex rolls 11 on the d12

Does this count as:
Alex's dice stand with 11/6
Charlie and Bert pick up their dice, Charlie goes second, Bert third.

Or does this count as:
Charlie's dice stand with 7/5
Alex and Bert pick up their dice, Alex goes second, Bert third.

Basically, does re-rolling a tie among the losers give a chance to overtake the leader?

I don't think it matters, as long as you do it always the same way. After all, either way produces a perfectly valid result.

The very, very slightly more practical ruling is to have the initial high roll win and stand, and let the initial losers reroll against each other only to establish who goes second and third. Their rerolls don't let them overtake the leader. So if you need a ruling, there you go.

(Rationale: either way, occasionally you'll get a second tie and have to reroll again to resolve it. However, if you do it the other way - if you compare the rerolls to the original high roll too, not just to each other - the second-tie circumstance will come up, like, three times per zillion instead of one time per zillion. When I say very, very slightly, I mean lord, very slightly.)

-Vincent