I don't have any specific information about rooting mulberry cuttings, but if they root the way most other plants do, then this isn't a good time of year to take cuttings. Since you're in Oakland, I wouldn't imagine mulberries would be making new growth right now, so the youngest cutting you could take, from last year, would still be old. So I'd take the tenderest looking little piece I could find. I normally like to cut a sprig of new growth in the early summer, before it's hardened off. I try to get a good "heel" piece, that is, one that's wider at the bottom where it meets the limb than it is for the length of the cutting. I like to get a piece that's a little thicker than a straw, no thicker than a pencil. I immediately put the cutting in water, then trim off all but the top leaves. I "wound" the cutting at each leaf node by cutting into it deep enough to expose some of the inner part of the twig, but certainly not too deep. I also make cross cuts at the very base of the cutting - plants root better when they've been cut into a little. Then I stick a pencil or stick into some wet potting soil leaving a hole for the cutting, put the cutting into it, water it again to make the soil adhere around the cutting, then put it in a warm, humid place in the shade. I keep it moist but not soggy. Various plants take various lengths of time to root, some never root at all.
I got a cutting of a passionvine that I really wanted in Florida in early winter, and it was way too cold root it outside. So I made up a rig on my stove in an old canning pot with water on the bottom that I'd heat periodically to give the cutting the heat and humidity it needed, and it worked! I now have a good rooting cutting of this passionvine - my husband really laughed at this!
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Sherry

Mulberries are often propagated by 'truncheons' - very large cuttings, sections of branch 1-2m long and 5-10cm diameter, buried for 3/4 of their length in the ground. This would however be impractical (and probably illegal) to transport to Canada. Also cutting-grown trees are typically less healthy and less long-lived than seedling-grown trees.

Probably simplest just to buy a new one at a nursery in Canada - they should be readily available. If wanted for fruit production, Black Mulberry (Morus nigra) and Red Mulberry (M. rubra) are better than White Mulberry (M. alba).

Resin

I know this response is years too late for the original poster, however for new questers one can take truncheons or cuttings from mulberries in the summer (when the soil is 70 deg F or higher), and stick the cuttings and truncheons directly into the ground. If taking truncheons, you probably don't need to water so long as you get 1 inch per week, smaller cutting would need to be kept wet since they'll only poke a few inches into the ground.

Yes truncheons and cuttings are not as long lived, but mulberries from seed wont bear fruit for 10 years, while truncheons sometimes can bear fruit the very next year. It might be an excellent idea to plant both so you have some producing early, and other longer lived from seed trees coming into production later on.

I am making a hedge from red mulberries about 1200ft long. I am currently transplanting seedlings and anything small enough to transplant healthily, as well as truncheons and sowing seeds starting this summer.