I recommend checking with the system manufacturer about possible fixes in BIOS updates or the cooling solution.
I do have the exact same issue with the exact same notebook.
ASUS X550LN, Core i5 4210U. Bios version 305 (there is 306, but it doesn't point to a fix of that).
Processor runs fine at 2300-2400MHZ when system is booted. When I start converting an HD video, clock goes down to 800MHZ. I haven't discovered what to do so far. Have you?
The problem might not have been the CPU, but instead the GPU.
Lots of people have reported the same or similar problems with their laptop. The solution might be in the newest nvidia drivers (353.30).
I'm going to test it out and reply if it fixes the problem.
Yup, seems to have fixed it as I had no significant usage or frame drops. The temperatures were on average 69 for the CPU and 64 for the GPU (not representative as it is quiet hot where I live).
I recommend doing a clean install of the driver, using DDU to completely remove the previous ones. If the newest driver doesn't work rolling back to the 344.xx drivers might help but at the cost of performance.