Does accomplishments progress carry across multiple playthroughs?

Mass Effect 3 has several "Do X Y times" accomplishments for example, the "Bruiser" accomplishment which requires a certain number of melee kills.

Do these accomplishments require me to complete all of these action on one playthorugh or does it stack between multiple saves? If progress does carry over between playthoughs, do I need to import a save into a new game for progress to carry over or can I start an entirely new save?

Yes.

All Mass Effect 3 Achievements carry across playthroughs and profiles and multiplayer.

(And apparently Warp counts as setting enemies on fire? Who knew?)

Achievements do carry across play throughs and saves.

Achievements also carry across deaths and reloads. I had to reload the mission where you storm the camp and I killed a brute while it was charging at me to get that achievement. The game glitched, I reloaded, but I retained the achievement.