Someone once asked the question, if a person slew another person in a city of refuge what would happen? Would they be forced to flee to another city of refuge? Using the text I believe there is a simple answer. The explanations of the city of refuge are in Num ch. 35, Deut 4:41-43, and Deut ch.19.
The city of refuge is a safe place for someone that accidentally killed another person to hide from a relative of the victim that wanted immediate vengeance. The slayer has to stay in a city of refuge until the presiding high priest dies. If they leave beforehand they are not protected from an avenger. There are only 6 cities of refuge initially, 3 east of the Jordan River and 3 west of it. The Israelites are allowed to add 3 more if their territory grows as a result of God’s blessings (Deut 19:8-9). Numbers 35:6 says that six of the cities that are set apart for the Levites, should become cities of refuge. This means these cities were unique and only the Levites live in them. Furthermore, Numbers 35:25 says the community was responsible for making sure a person was safely escorted to a city of refuge and Deut 19:3 says that they need to set up these cities in good locations that everyone can reach with easy roads to travel to each city.
The situation described would not have happened unless a Levite slays another Levite, or a refugee slays someone while in the city of refuge. The Levites are the tribe that works with the Tabernacle/Temple and they aren’t even allowed to serve in the military so I don’t think they can commit vengeance executions. That is why their cities are the cities of the refugee because it is supposed to be set apart as holy and free from violence. Based on what is written, if a Levite slays another Levite, they would just go straight to trial and if it was not an accident they can simply have them executed for murder. If it was an accident since they are all Levites and are supposed t be holy there would be an understanding that they cannot avenge the death. If a refugee who is already in the city for a slaying, accidentally lays a person within the city of refugee I imagine that trial would happen only if he is found not guilty in the preceding trial for the accident that brought him to the city in the first place. If he is found not guilty in both trials he lives and stays untouched until the high priest dies. I don’t believe the Levites can avenge their family members who were accidentally slain since again they are set apart as holy and didn’t even serve in the military.