If a flyer with running shot is carrying someone with the Shield team ability, can they move, have their damage increased by the Shield member, and then shoot?

No. The SHIELD TA must be activated *during* the attack action; it can't be turned on before the Running Shot begins.

And since a figure can't be given a non-free action after being taxied, your only option is to have the SHIELD already in position next to where ever the Running Shot figure will end up.

Thanks Normalview, that's what I thought.

Couldn't perform the Shield action than do the running shot? Assuming the Shield character is in range at the time of the action.

Also, what is their range for this action?

No, you can't activate the SHIELD TA before the Running Shot. As I said above, the SHIELD TA activates *during* the ranged attack. The ranged attack doesn't occur until after the movement part of Running Shot is over.

And there isn't any range to the SHIELD TA action. The attacking figure must be adjacent for the SHIELD TA member for the TA to be used.

I prefer three adjacent Hydra agents in the fore with an E/V Hawkeye + Trick Shot taking up the rear (+3 AV), but I digress... The SHIELD TA can be effective when played wisely, but for the most part, it remains widely underused. I'm hoping WK comes up with an ICWO-equivalent for SHIELD members that lets them use Fury's dial!