I have been having this problem with a variety of games. The ones I can recall specifically right now are Life is Strange and the one I was just playing, The Old City.
Both games, and at least one other I can't remember, play fine for the first ten minutes or so. Life is Strange is getting about 80fps, and The Old City is just over 60fps. But after a little while, both games drop to 22fps. I thought it was just specific to Life is Strange initially, and beat that entire game, but that was about a month ago and now I'm having the same problem with yet another game. What could cause it and how could I fix it?
I've tried stopping as many other processes as possible, turning off every video game and streaming overlay I can think of, and just now I tried going into the BaseEngine.ini file and increasing the PoolSize value from 160 to 2048, with I believe corresponds to MB of GPU memory. My rig is an AMD FX-8350, 16GB of RAM, an nVidia GTX 760 with 4GB of RAM, and 64-bit Windows 7 Home Premium. I'm using an ASUS VG248QE 144Hz monitor.