The only games I've tested this with were Flash games, but like I said, it also occurs with Bandicam.
I really can't find the pattern here, but I can tell you that this all started while recording this video (though I don't think you'll notice it very well as, at the time, it wasn't as bad):
https://www.youtube.com/watch?v=rsNyXYqbUPU (Not to advertise, lol, but thought it could be helpful to know when it started)At some point while recording the level with those dark clouds, Bandicam started behaving a bit strangely to me; it wouldn't lag while I recorded, but it would take MUCH longer to actually start recording after already clicking the record button, sometimes the same way when ending the recording. SOMETIMES.
And eventually, this behavior started sinking in Super Mario 63's performance, along with Last Legacy. This "lag" would occur when the Shine Sprite(s) would begin to activate. BUT. Eventually this started to affect loading times--NOT Super Mario 63's infamous "loading times," but anything that requires loading (maybe something like what Runouw said about taking a long time loading a relocated source of memory... or whatever it was he said.)
FYI, this problem DOES NOT affect web browsers. Also, I haven't experienced this with videos yet (which makes me happy).