In case of "Relics", we not only can but probably must think in terms of major cuts in the action. Our heroes are buzzing inside and around an object the size of a small star system, without the benefit of warp drive; their antics by necessity must take several minutes even when mere seconds are depicted on screen.
In practice, then, there's probably quite a bit of action between LaForge's hail to the E-D and the bit where Scotty says the transport ship can't maintain shields much longer, even if these are cut back to back. Picard and pals then discuss that they have one minute and forty seconds till they reach the doors; the camera cuts to LaForge and Scotty for a few seconds; and Picard orders the transporter beam energized. Most of those hundred seconds went "unused", and it's not difficult to assume Picard used them to give LaForge instructions to drop shields on his mark, then have them automatically re-raised to keep the Dyson sphere door open a few extra seconds.
Of course, the other possibility is that LaForge and Scotty were beamed to safety at the exact moment the shields of the transport ship collapsed - that is, when the torpedoes from the E-D hit her. That should be quite doable: the rescue of Chakotay in "Caretaker" is similar split second stuff, for example. Picard simply commanded "energize" a few seconds in advance, because transporters (and transporter chiefs!) typically require a bit of lead time...
Whether Janeway's shields were up in "Caretaker", "Maneuvers" and "Dragon's Teeth" which all involved transporter rescue during combat... Well, it is debatable. It would be quite possible to drop shields for a few seconds, and quite possible for the camera and the audience to miss the relevant commands. Delta quadrant weapons, especially Kazon ones, were always depicted as somewhat wussy, and might well be allowed to hit an unshielded starship a few times.
Timo Saloniemi