I’m a little late to the party, but thought I’d throw my 2 cents in.
I’ve always found mulberry can be a little persnickety to graft. Here are a couple of things I’ve found helpful. As always, your mileage may vary.
I like rootstock to be at least somewhat awake. I don’t graft totally dormant stock as one would apples or pears. Similarly, I don’t mind if the scions are a bit farther along. If the buds are swollen and showing a bit of green, that’s a plus in my book. My reasoning is, I want the scion to wake up at the same time the stock is pushing sap, that way the whole works is ready to hit the ground running.
I prefer a side graft part way up the rootstock. The side graft lets you work with a range of scion and rootstock sizes, so there’s no need to match calipers, at least within reason. If it’s typical 1/4” or so rootstock, I’ll do a side graft roughly halfway up leaving the upper part of the rootstock, but pruning it back heavily. I like the sap to push PAST the graft Union, and continue flowing, as I’ve know mulberries to otherwise seemingly abort the top and instead push from dormant buds further down near the crown. By pruning some of the remaining stem, there are places for excess sap to weep out, which helps with flooding of the graft Union. When the scion does push, you have a ready made stake to secure the growing scion until the graft Union develops some strength.
Someone mentioned the merits of rooting vs grafting. IME, varieties vary greatly in how easily they’ll root. Many feral albas root extremely easily, akin to willows. A lot of the cultivars are at least somewhat harder, with some, like Gerardi and IE being downright hard unless you take them at just the right time. The best time (and only time for the tricky ones) is when the new growth is firm but not yet woody. Mid-July is usually about right in VT.
In a pinch, I’ve had decent luck root grafting. Get some ropey roots and cleft or whip graft ‘em right on. It’s a little tricky, because the root bark is quite different than what you find on the stem. I don’t know for sure, but my hunch is that to want the root piece a tad oversized, as the root bark is quite a bit thicker. One other trick I’ve had luck with is to graft onto a cutting of variety that is east to root. I like the contorted variety, ‘Unryu’ for this as it roots very reliably. It also makes super long ropey roots that are perfect for root grafting.
Any way you slice it, mulberry grafting is somewhat harder than gateway stuff like apples. It’s still pretty low hanging fruit (pun!) though compared to truly tricky stuff like nuts and conifers. Expect to fail some and play around with it. You’ll be pleasantly surprised sooner or later!