Mouth rinses. There are different mouthwashes for different purposes. Some need prescriptions while most don't. I am not sure what you are using or why (or if you even have to use it in the first place). Whatever the reason, I can't see why you would want to limit the effectiveness (if it is effective even) by rinsing out. Your own dentist can best answer your question and discuss this issue with you.
You Should Rinse. Because the effects are applied during the swishing of the mouthwash, which is why the recommendation to swish for 30-60 seconds. Some mouthwash also can leave stain in your teeth, Chlorhexidine is a good example.