Potentially getting a new monitor, question about refresh rate?

I normally use a TV for my gaming PC, but unfortunatley it seems to be on its last legs so I've found a monitor (1080p@60Hz) that's not too expensive that I'll probably be switching to.

I have a Nintendo Swith that I sometimes use on my TV so I was thinking about getting a HDMI switch so I can have both my Switch and PC plugged in (and I don't have too mess around with cables lmao) but after looking at HDMI switches, it says it supports 1080p@60Hz - so does that mean if my new monitor runs at 1080p@100Hz and i then run that through the HDMI switch (1080p@60Hz), would it cut down my refresh rate?

I'm sorry if this is kind of a dumb question, but It's honestly the first time I've really thought about refresh rate and a lot of this stuff kind of goes over my head lmao

Thanks!