I set the freshly-filled containers in a basin of water. You may need to adjust the level so that your containers don't go floating off and capsize!
The first soaking is likely to take several hours. Don't go less than 2. Then, you need to drain it - tilting at several angles and leaving it for an hour will help. That draining is important if you want to bring the container into the kitchen, sow the seed, sprinkle it with a little more soil, and set it somewhere warm for the next several days.
Everything is bottom-watered for weeks & weeks. My basin is one of those that you buy at Home Depot for mixing concrete and such. It is made to fit inside a wheelbarrow and a standard flat fits nicely in there.
I use a an old dish pan to wet my soilless mix before making my soil blocks which is my favorite way of starting seeds... I still use cell packs for some and 3 inch pots for others but always pre wet everything and bottom water til I transplant to the garden...
I fully saturate my newly mixed raised bed soil. Otherwise, it just skims off the top. So a 6x6ft, 4in deep bed is a mix of 50# sand, 3cft peat moss & 50-100# soil. I dump it all in trying to scatter it in a way mixing will be easy, then mix it all and lastly, turn on the hose full blast. Spray the whole thing for 5min or about 30 gallons of water and then put the hose down IN the bed to keep filling. Then, with my muck boots on, I just stomp around until everything is wet, no puffs of dry dirt or anything, and there are a lot of hidden pockets of dry dirt in there.
I then wait about 3 days before planting to let it drain out all the extra water. Soil will now equally absorb water and I don't have any annoying dry spots.
I thoroughly soak potting soil before I use it, if the bag didn't get rained on, I put some in a bucket, fill with water and mix it. Then drain extra water, let it dry a bit and then use it to plant.