As a start, it is highly unlikely that the plants will die from this.
This is something that I've seen very frequently with members of the mint family and shouldn't be a huge concern. Growing up, our various mints would get something like this on occasion and they still grew like weeds year in and year out.
On the one hand, I don't think the exact cause of the brown leaves is known, but it is usually associated with a highly stressed plant that is trying to protect itself. Some of the leaves look like they might have been attacked by a pest and others look like it's just the plant reacting to a potential infection vector. If you recently got a surprising wet surge in your weather, it might have given various bacteria a boost in strength, or with the dead heading it might have decided to put energy into new shoots. However, one thing that I'd be a little bit careful of is root rot, as you mentioned the rain. If your soil isn't draining properly it can also cause symptoms akin to this.
If you want a bit more reading about salvias in general, this has some excellent information.