Information urgently required: Switching work permit from Vancouver to Montreal

[basf]

Hi there,

I'm currently in Vancouver, with a work permit linked to a company here. I've been offered a role with a different company in Montreal.
They claim that they can apply to have my current work permit switched over to them and that I can work for them while the paperwork is being processed.
This doesn't sound right to me... can anyone confirm if what they are saying is true?

Thanks very much.