Let me use Montreal, Quebec as an example (Assuming you wanna leave Quebec), as I don't know which province you are in. First, you inform the Social Welfare and your lawyer (If you intend to retain his/her services) about your intentions to move and when you are likely to move. Then the office of social welfare will close your file with them since you are gonna open in another province. Then you move. When you arrive at your new province, report back to the office of their social welfare which might have another name and then they open a new file for you and probably put you in a home or at a shelter pending the time you get your apartment. Then you will inform the IRB in the province you are in of your new address and also Canada immigration. Mind you, if you are staying at a shelter you must give them the temporary address pending the time you get an apartment and then you inform them again of the new changes. (I bet you don't wanna miss any mail)
If you intend to take your case file to the new province, just retain the service of a new lawyer who will inform Legal aid and IRB as your new solicitor and he or she will be in correspondence with your old lawyer. Mind you some lawyers don't have that time and it may require you to meet your old lawyer and get your file containing your BOC and handing them over to your new lawyer.
I hope I made some sense here.