A ship's shields are down and it takes 3 crits. The ship next to it is going to use Draw Their Fire. Does the ship take the first crit away without the card being turned up? Do all three get dealt up and the ship chooses which one it will take?

You choose when it goes off. Before your resolve any ONE of the dice, you may use DTF.

3 crits.

You may use DTF before resolving the first crit, or you may deal a faceup damage to the target ship.

You may then use DTF before resolving the second crit, or you can deal the second faceup damage to the target ship.

You may then use DTF before the 3rd crit, or you can deal the third faceup damage to the target ship.

In each case, choose to use or not use DTF before resoling the individual die result, so you can't know what the crit will be before you choose.

Thanks.

Although there is only one ship that could really use the situation it is important to know that DtF is used while the "crit
is still in the "suffer" state and long before cards are drawn. Why is this important? Imagine a ship with DtF and Determination (possibly with an A-Wing if highly impractical) that if it knew an ally would take a face-up Pilot card could instead choose to suffer that result and immediately discard it due to Determination. It doesn't work too well on an A-Wing because it takes so few cards to kill but imagine if it was on something like Chewbacca who may not be a bad DtF holder to start with.

It also matters because if DtF waited for the card to be dealt that would mean the ship using DtF would have to suffer that CARD instead. You wouldn't be using DtF to draw a potential critical and have it take away a shield token instead which you may be able to get back later.

Notable is that you can only take one of the crits for the other ship... Draw Their Fire only has a single trigger, when the friendly ship is hit by the attack. You just suffer one of the results for them. As @InquisitorM stated, using your example, it can be the first, second, or third crit... but only one of them. The rest go into the intended target (unless there are other damage mitigation cards you might have). Also, as stated above, you don't get to see the crit before taking it... so you won't know if it's something potentially disastrous (like Major Explosion) or a nuisance (like Shaken Pilot) until after you make your decision.

Also if you think about it, assuming you have no shields left, it doesn't matter which of the 3 crits you DTF, as you will be turning the next two damage cards up no matter the order.

The only situation that might cause some thinking is if you flip the first two crits and they aren't that bad so you decide to eat the 3rd crit as well instead of using DTF to give it to your other ship.

I have seen the (accidental) attempt to misuse DTF and/or Selflessness enough times to feel this extra bit is warranted: The attack hits the defender, even if the defender suffers zero damage. It is only after the attack has been determined to hit that a player may trigger and resolve DTF/Selflessness.

For Example: In the original question, if the defending player has 3 friendly ships at Range 1 all with DTF each one of those ships could cancel a different critical hit result (before revealing the damage card) and thus allow the defender to remain undamaged by the attack. Despite being undamaged, DTF can only resolve after an attack has been determined to hit. Therefore, in this case, should a post-attack-hitting effect need to be resolved against the defender it still is resolved (such as assigning the Harpooned! condition card or resolving the Shadow Caster title).

I've found this to be a relatively common point of confusion, likely caused by act of "cancelling" attack dice...