My understanding is that small, shallow wounds that are relatively clean with minimal tissue destruction can be cleaned and closed. If you miss some dirt, it happens, the wound will swell and hurt. In which case you cut the closures, drain, irrigate, pick out any debris that got missed the first time and pack.

Deeper and larger wounds, particularly ones where chunks of meat are missing or mulched, usually shouldn't be closed immediately. They should be cleaned, debrided and packed. Used to be packed with betadine soaked gauze but there are more modern materials now. The packing is removed, the wound gently cleaned and repacked regularly. This is designed to allow the injury to heal from the inside out.