A line of credit charges interest on the balance you carry each day. So the day a draw posts is the day the meter starts running. A draw just means money you pull off the line of credit. When you're sending one chunk at the mortgage, the difference between a draw that posts today and one that posts Thursday is small. Do it twelve times a year and it stops being small.
An online transfer to your own checking at the same bank is the fastest, and it's the one to use by default. It usually posts the same day, sometimes right away, and the money is spendable immediately. If your line of credit and your checking account are at different banks, that transfer runs as an ACH, which is the electronic network banks use to move money between each other. An ACH takes one to three business days, and the draw usually posts the day the money leaves rather than the day it arrives.
The convenience checks are the slowest. You write one, the person you paid deposits it whenever they get around to it, and it can take several days to come back and post as a draw. That creates two problems. You don't control the deposit date, so you don't really know your own balance, and some lenders charge differently for check draws. They're useful for paying somebody who won't take an electronic payment. Not for much else.
The card is fine for purchases and it's a trap for cash. A purchase posts as a draw right away, with no fee on most lines of credit. Pulling cash at an ATM with that same card can get coded as a cash advance instead. A cash advance carries its own fee and sometimes a higher rate, depending on how the lender set the product up. Ask before you use it that way once, not after.
A wire is the one for a large chunk headed to a mortgage servicer or a title company. It lands the same day if you make the bank's cutoff time, it usually costs a fee in the twenties or thirties, and the money is final when it arrives. For a $40,000 chunk, paying $30 to have it land today instead of Thursday is a good trade. It also removes the risk of the payment posting in the wrong month.
Whatever route you use, the second half of the job is telling the servicer what to do with the money. A payment that shows up with no instructions can get applied to next month's payment. It can land in a suspense account, which means a holding bucket where the servicer parks money it doesn't know what to do with. Or it can get split across principal, interest, and escrow the way a regular payment is. Some servicers have a principal-only box in the online portal. Some want a separate check. Some want written instructions. Confirm the money posted to principal before you send another one. The whole point of a chunk is knocking down the principal, and a chunk sitting in a suspense account isn't knocking down anything.
One habit that saves real money. Don't draw until the day you're sending the money. Drawing on the first because you plan to pay the mortgage on the tenth costs you nine days of interest on money that sat in checking doing nothing. Why those nine days cost what they cost is in daily interest, monthly interest, and why the difference runs this strategy.