Can shields be raised at warp?

Question

Sometimes in Star Trek, ships are seen firing at warp in a high-speed chase. One example was Star Trek: Into Darkness, when the Vengeance fired phasers on the Enterprise at warp, causing catastrophic damage. Another example was in DS9, when a runabout fired phasers on a Dominion fighter, destroying it. Similarly, in Star Trek: Voyager, the U.S.S. Voyager fired one photon torpedo at the U.S.S. Equinox at warp, causing heavy damage and forcing the Equinox to exit warp.

However, in the Star Trek: Voyager (which, I know, I know, is notorious for its discontinuity) episode "Message in a Bottle," the Prometheus (split into three parts) pursued a nebula-class starship (perhaps the Sutherland?) and fired on it. This time, however, the Nebula-class starship's shields are clearly up. Although it does sustain heavy damage, its hull isn't breached and torn open like in the other cases. This indicates that either shields can be raised at warp, or they weren't at warp, and I'm just confused.
Is it possible to raise shields during warp? If so, why doesn't it happen more often?

Answer 1

This answer, by necessity, includes some out-of-universe considerations.

Although various tie-in comics (such as Countdown) were produced in an attempt to stitch together the continuities, the Chris-Pine-helmed Star Trek (2009) and sequels were effectively reboots of the franchise. They are, essentially, alternate universes to and incompatible with the "main" Trek universe as seen in TOS, TNG, DS9, and Voyager. The recent series such as Discovery and Strange New Worlds make that incompatibility especially clear.

But, to the point: shields at warp were commonplace prior to Star Trek (2009). (You cite some of the occurrences, in fact.) "No shields at warp" was a new (and, in my opinion, pretty unusual) invention of, and generally confined to, those three movies.

(To clarify two of your examples: the DS9 Episode "Treachery, Faith, and the Great River" saw Odo and a Weyoun destroy a Jem Hadar fighter while at warp. The runabout takes several hits at warp and Odo/the computer even has dialogue about shield strength during the scene. Weyoun specifically provided tactical knowledge that enabled Odo to punch through the Jem Hadar fighter's shields, and shields were shown during the effects sequence as they did so. In Equinox, the U.S.S. Equinox itself was itself in very bad shape after years of barely scraping their way across the Delta Quadrant. It is quite plausible that the Equinox was simply just so badly damaged that one shot was enough to knock systems offline.)