I can't speak for doctors - their rules are a bit tighter, but I can speak for lawyers.
Back in the 1980s there was a move to open lawyer mobility - ie the right to practice in another jurisdiction. It opened up a lot and the system was negotiated across jurisdictions by the Federation of Law Societies of Canada. For example, a lawyer from any common law province can practice in Manitoba for 100 days in the year if he can provide certain assurances including being a member in good standing of his home law society, having adequate liability and defalcation insurance, the practice is temporary in nature and no economic nexus to Manitoba is created etc. There are numerous conditions that apply but to answer your question, the Professional Code of the host jurisdiction will govern but, in general, complaints will be handled by the home jurisdiction with the consultation and cooperation of the host jurisdiction.
Quebec is an issue primarily because much of the law is civil rather than common and because of the language requirements.
Professions in general have been moving to easing age-old restrictions on trans-jurisdictional practices in temporary situations. If the mobility is more permanent then a transfer to the new jurisdiction is required. When I permanently moved from Manitoba to Ontario I transferred my practicing certificate and the process was straightforward and easy.