Since your babies are still growing, consider going up one pot size each time the roots are coming out the bottom of the pot they are in.
Here's what I do when I start tree seedlings in peat pots.
I select a pot that is two inches larger in diameter than the peat pot. I fill the bottom of the pot with a good potting soil. I like Miracle grow because it has fertilizer in it already and I won't have to add extra fertilizer.
I carefully tear off the top rim of the peat pot and the bottom of the pot.
I nestle the seedling into the pot and back fill with soil even with the top of the soil. Carefully pack the soil around the root ball. I then use a shower spray to settle the soil. If the potting mix is lower than the root ball, I add more potting soil until it reaches the same level as the seedling.
Your little tree will now be ready to take off growing again.
If you want to prepare your tree for outside, set it out in a shady spot for one hour the first two or three days. Over the week, increase that time outside by one hour until your tree has been outside 5 hours. Then begin putting your tree in part sun for an hour at a time.
I prefer to leave my young trees in part sun/part shade until the tree is several season old.
Lime trees and any citrus tree is not cold hardy and will need to be brought in before a freeze and placed in a sunny window for the winter. You could provide it with a grow light.
