easiest way to fill cell trays with media?

I have a bunch of 144 cell trays that I use for seeds and shrub propagation. They have a large hole in the bottom, so each time I want to use them, I place small, cut squares of paper towel in each cell, fill with media (perlite/peat moss mix), press them down, and refill as needed. This gets really tiring for several trays. Does anyone have any suggestions to speed this up? Putting in the dry mix without the paper liner causes the mix to spill out at the start. Using moist media and a finger in the bottom of each hole keeps it there until I water from the bottom. If I could find some long sphagnum moss, I would throw some of that in the bottom, but I just have peat moss of mixed lengths. 1/2 of that would flow out, but the longer stuff may help plug in the regular media above it. Has anyone found a way to quickly fill these and keep the soil-less mixture in? I looked into bio-sponges, but they wouldn't fit and were quite expensive even on sale, compared to good potting mix.
Thanks. If you can think of another forum that would work better for this, please let me know.

What if you started the cells on a cookie sheet that would seal the bottom and allow you to water. The flat surface would hold the mixture in and the small rim would allow you to water from the bottom and "seal" each cell.

Unless you have larger than normal holes in the bottom of your tray, I would say your mix is packed too loosely. I use 128 plug flats with 1/4" holes in the bottom for most of my seed starting and I never have any problems with bottom watering. I use media that is extra fine and made for use with plug flats. I fill trays with moist media, tap several times on a flat surface and then mound more media on top and "rub" it in with my hand. I sweep the extra media off with the side of my hand and plant. I water using a standard 1020 flat.

Many people are afraid of compacting their media and not having seeds/roots be able to penetrate the mix. I've found that it is almost imposible to compact planting media to the degree that plants won't germinate and grow. In fact, many people leave their media way to loose.

I take a clean bucket, add in the medium, water, and mix with my hands until nice & moist. Then place directly into the seed trays.

I will give those a try. I'm glad to hear what is working for other people, and hadn't expected the media to stay in that well. Thanks so much!

I was sent some trays on ebay, and am using them with clear plastic bags. I lay the bag out and put the tray on it, pack the trays with dirt, and shake the precious, expensive medium back in the bag it came in, then seed the tray and put the clear garbage bag around the seed starting tray and add a couple of quarts of water, laying all on a table on our porch.