Spells only get the benefit of buffs that are up when the spells are cast. So if you put all your DoTs up on a target and then use a spell damage trinket, the DoTs will not get a bonus from that trinket. Likewise, a shadowbolt will not get the buff from a trinket or whatnot if the buff runs out before the shadowbolt is cast. However, this does mean that if you cast a channeled spell (drain life, arcane missiles, mind flay) while a buff is up, and the buff runs out in the middle of the channel, any damage done after the buff runs out is still getting the buff, as the spell was cast while the buff was up.

To add to Nizari's explanation, with Haste procs/trinkets you gain the lower cast time when you start the cast. So if a Haste proc/trinket wears off during a cast, the cast time will not suddenly increase.

It's an interesting mechanic actually.
For example, I raid as a shadow priest with Dying Curse and Sundial of the Exiled which both have hefty spellpower procs. When either procs (or both in the same window) I make it a point to recast Shadow Word: Pain as it is refreshed by Mind Flays because of Pain and Suffereing. When refreshed the SW:P maintains the bonus damage from the procs which have since expired.

I'm not sure if this is supposed to work in this fashion, or if it occurs with any other spells right now, but it's worth noting.

@Segrada. Not sure if it's intended or not, but this is definitely the way to play a shadow priest. In fact, if you go to Elitist Jerks there is an add-on linked and explained there called Project Greenlight. It basically does one thing, which is monitor various procs and raid buffs for SP and crit and will then tell you the best time to refresh your SW:P.