I have a permit for 3 years until 2021, tied to my current employer. In order to work with a new company we need to make a process to change the name in the permit. I have the paperwork ready because I made this many times in the past.
From the company: I only need the offer/contract and some other information to fill the forms. The government fees are not expensive (aprox. $400 - $500 CAD)
To ensure that this process is right, I usually work with a lawyer.
This process could be in the border (one day) or on-line (three weeks). The on-line process is sometimes better so we have time to review the information with the lawyers and the immigration officers. But the process in the border is faster. So, it dependes on the new employer if they can wait or either if they prefer to do it in the border. On-line process is usually better.
Canada renegotiated the trade deal with the United States and Mexico last year. Before, this agreement was called "North American Free Trade Agreement" (NAFTA), now the name is "U.S.-Mexico-Canada trade agreement" but the rules for the working visa are the same as before.