Pitchers during benches clearing incidents.

Hi, so I had a baseball question and I tried posting this in one of those baseball for beginner communities and it got removed for some reason. So I will try it here. Sorry if this is the wrong place for this.
My question was why during benches clearing brawls do the pitchers from the bullpens run all the way to Home plate and the mound when nothing usually happens. I was thinking about this because of the Red Sox and nationals last week.
On the one hand, I get it because you are showing you are supporting your teammate and you have their back. But usually everyone is standing around. You are risking an ejection, fine, suspension, or injury. And adding more people to the situation could escalate it instead of making things easier.
Is there a rule somewhere that says every player in the bullpen and bench must enter the field when situations occur? Could a couple relievers be like no, I’ll sit this one out and not get ejected?
This has confuse me for a minute.