I don't use a bin at all for my cuttings. Just a cup on top of a cup, each have ~12 holes melted in them. Once they have enough roots to support the leaves I slide the top cup over to the side about 1-2 cm and refasten with tape. After a day or two like that I take the top cup off. The air where I live is very dry. They stay in the cup until at least 70% of the cup has roots in every square inch. I also want to say that least 1/3 of the roots are relatively mature regardless of caliber(dark tan to brown but not dead brown). I don't fertilize in cup so I can't let them get too large.
I use the same mix I use for my larger potted trees when I go from cup to pot. The soil is very damp but not soggy, I wouldn't be able to squeeze moisture from it. Put enough mix in the bottom of the pot so that when I set the cupped plant in it is up to the right level. Then I spoon the "soil" all around the cup until about 3/4 up to the top of the cup. Lightly pat/slap the side of the pot all around to settle to soil around the cup. Add more soil to bring back up to the 3/4 mark and pat/slap again. Then grabbing the lip of the cup I lightly wiggle the cup in all directions to slightly enlarge the void my cup is filling. When I lift the cup out the shape of the void remains and doesn't collapse.
I find at least half of the time I have had to use a butter knife to coax the roots off of the cup, the first time I did this I thought it was a disaster and the growing medium in the cup crumbled when I inverted the cup. But I do this over a plate and try to be gentle with the crubled root mass. Place the root ball in the void created in the pot. If it crumbled, glently attempt to use something like a chop stick to hold some roots up near their original positoin while putting the cup medium into the void with the roots. If the medium is dry misting the exposed roots before adding back medium is a good idea. Layering a bit of the new soil with the cup medium as you fill the void works great. In the end the root ball is covered with new "soil". Do the tap/slap one more time, I then put a layer of perlite about a centimeter or two thick on top of the soil to help with moisture control, this was suggested by Jason and thus far seems to work well.
Using a spray bottle I mist any debris off of the leaves(if no debris mist anyway), and water the entire surface with the mister until I am satisfied a have added a good deal more moisture, but not enough that it has run out the bottom of the pot. Place near a window not in direct sun for a day or two, then move to direct sun. Within a week the plants explode with happiness and growth. I haven't lost one yet.
If in the first day I experienced severe drooping or wilting I would use a baggy or humidity dome of sorts along with misting until the plant recovered, keeping it out of direct sunlight. Thus far I haven't had to do this with up-potted cuttings. But I did have to do this last fall with nearly all the plants I received in the mail.
Hope this is helpful to someone.