The RCMP AFAIK do not have a mounted contingent anywhere. As for sled dogs, the last few decades, they’ve been replaced by snowmobiles. For chaising anyone violating some laws about off-road vehicles, etc they may also use quads. One contingent tested a Hummer for on/off road, but they don’t do too well in swamp and mud.
The RCMP provide police services in small towns, remote areas, federal lands (airports, etc.) as well as highway patrol and provincial police in any province that does not have a provincial police force. (The Ontario Provincial Police, Surete de Quebec, Newfoundland Constabulary, etc. ) The RCMP are especially good at tasering to death confused Polish immigrants and then lying through their teeth on the stand.
Most recruits can expect to spend their early years in remote locations like the north, or native reserves. They handle anything from traffic offenses to domestics to burglaries to murders. Later they can bid on fancier jobs down south, or at international airports.
I remember once shutting down a whole wing of the Calgary airport for 10 minutes until the RCMP showed up, many years ago. Ninja throwing stars are illegal weapons in canada. When the trained monkeys fo the private security service find one, they have to shut all the doors and call the RCMP as it it were something truly lethal like a pistol, or bomb (can I say that at an airport?). So we stood around waiting until the RCMP came, and based on the Xray, they opened my carry-on and pulled out the ornamental glass star, laughed, and sent me on my way… Looong before 9-11.
There are additional wings of the RCMP that do white-collar crime, will investigate other police if asked, do investigations across provincial boundaries, etc. There is CSIS, a separate group whose mandate is to mess up possibly international espionage in a uniquely Canadian way; sort of like the FBI arresting Moussaoui and missing the big picture. And CSE, our also Canadian flavour of the NSA. But anything criminal and national, the RCMP has jurisdiction - biker gangs, organized crime, foreign terror groups recruiting locally (shared with CSIS), security fraud, etc. Of course, if it’s a provincial law, a local provincial force may have first dibs on the case…
All criminal law in Canada is under the criminal code, is federal. All peace officers are mandated under federal law, so they are all empowered across Canada, including the Podunk Police force. It is just simpler for a small town to get their one or two officers from the RCMP than to set up their own collection of Andy Griffith and Don Knotts. The police have serious training (RCMPis still a prestige job) and answer to a hgher authority in Ottawa, so they are relatively immune to the usual pressures of a small-town police force.
For example the East St. Paul, Manitoba, police force was recently dissolved over allegations (more than allegations) that they gave a drunk driver from the Winnipeg police force a break; deliberately lied and messed up the evidence so he could not be convicted of Impaired Driving Causing Death. Simpler to teach the country a lesson, and fire 11 police officers, and get the RCMP heirarchy to replace them. Maybe next time the local police will think twice. Most local police are poeple who could not egt into the RCMP academy.