My understanding is that Ontario and Quebec have their own police forces who have jurisdiction on their lands, the RCMP often patrol federal lands such as airports, however Toronto police still patrol the airport. Municipalities and cities often either have their own police or contract policing out to the provincial forces. Take Martensville SK which had its own force which fell into disrepute, they were disbanded and the RCMP were hired to replace them. Smaller municipaities do complain of the cost of RCMP policing and sometimes form their own force as well.
As for jurisdiction, to my knowledge even military police can make an arrest, but will hand the alleged offender over to local authorities. Where I live it is somewhat shared, Halifax Regional Police and RCMP both patrol, however HRM police mainly work within the urban areas of Halifax, Dartmouth, Bedford and such, while the RCMP patrol the highways.
Certain offenders may be handed over to the RCMP depending on the offense, but if your guy committed an offense in, say Mississauga, and was caught by the OPP on an outstanding warrant, he would likely be handed over to Peel Regional police despite any protestations that he wanted to deal with the RCMP. I guess what I'm saying, generally, is that no matter where he is caught he will eventually deal with the force in the jurisdiction where the alleged offense took place.
That being said, a person under arrest has the right to:
- be immediately informed of the right to silence, (very important)
- be immediately informed of the reason for the arrest
- be immediately informed of the right to counsel
- be promptly allowed to contact counsel once police have the situation under control and the person under arrest (R v. Strachan,  2 S.C.R. 980), not like 4 hours later.