I start up Starcraft 2 fine and get around 30-50fps but when the mission is over and a cinematic starts my fps drops to around 4-10 and stays like this until I exit out of the game and start it again, anyone else having this problem?

I don't have that specific problem,although when a mission starts I drop to about 10 and stutter for a good 5-7 seconds before it jumps up to 100fps 

when missions load mine jumps up in the hundreds will fast then will drop low again. 

Set your graphical settings to low... if they are already set to low, go find the video options and make sure the option to view cutscenes in at high quality is also set to low.

" I don't have that specific problem,although when a mission starts I drop to about 10 and stutter for a good 5-7 seconds before it jumps up to 100fps  "

There has been words of the game bringing systems to crashes since they overheat the graphics card.
 
@Blizzardsaid:

 Certain screens make your hardware work pretty hard

Screens that are light on detail may make your system overheat if cooling is overall insufficient. This is because the game has nothing to do so it is primarily just working on drawing the screen very quickly. A temporary workaround is to go to your Documents\StarCraft II Beta\variables.txt file and add these lines:

frameratecapglue=30

frameratecap=60

You may replace these numbers if you want to.

I also get the same issue, even though the game recommends I set everything to ultra. If I drop the shaders to high instead of ultra, then all cutscenes are super smooth. The game also accesses my hard drive like crazy causing the shittiest, stutteriest lag at the beginning of every game. Hell sometimes it happens right in the middle of a game and completely screws me up. I find it quite funny that a game that is visually pretty mediocre, can put so much strain on my system.

Check your temperatures

I think the game has issues with memory leaks in campaign , everyone I know has the same problems. The more you play usually the worse it gets.

" I don't have that specific problem,although when a mission starts I drop to about 10 and stutter for a good 5-7 seconds before it jumps up to 100fps  "

Don't seem to have this problem, even with my old school 9800gt
Hope they fix it, for those who experience problems

" @CharlieTuna:  If it were leaks it would probably slow down continuously during the longer period. But it drops down  from 40 to 1 FPS  almost instantly after just one mission is completed. "