Does a transporter using its move-and-attack power to make a close combat attack have to break away from its target following the attack to continue using its movement? I've always thought no, but don't see any wording in the rules that makes this any different from the Ultimate Falcon's "Flyby" SP.

No Transporters do not.
The character can move through squares adjacent to opposing characters.

No with the Move and Attack ability. The difference is you are making the attack mid-movement; you haven't officially ended your movement. With Falcon's Fly-By, it's Charge. You end your movement before making the attack. Thus, you need to roll a breakaway.

Likewise with Running Shot, you may use a carried character's Probability Control with RS and not with HSS, because they are placed down when the movement ends (occurs before the attack with RS, however not with HSS).

Ah, got it. The "Charge" part of Falcon's power is what makes the difference. Thanks!