Lei shen trinket and other intellect procs work fine. Dot's casted with these procs up will benefit from them for base duration of 15s and 18 s.
Problem arises with haste buffs. Haste is being recalculated during dot extensions. I. e. you cast VT and SW:P with LMG 30% haste buff (expecting 1 or 2 more ticks). Then haste buff falls off and shadowy apparition extends your dots. Tick period will be recalculated and your additional tick will be lost.
The positive thing about this is that it works in opposite way as well. You can cast DoT without haste buffs, then if u see shadowy apparition walking towards your target pop Power Infusion and your DoTs will gain that haste.
Because I am using Berserking, Power Infusion and LMG, I have decided to drop set bonuses for equal item lvl gear, because I think this may occasionally cause dps loss. This is also being debated on H2P forum linked by Drye.
PS : This conclusion comes from personal testing on live realm. My only pieces equiped were tier pieces so i had 10% haste, and was comparing tick periods between 10% and 116% haste using cd's at different times.