If you have both tanks set up and both tanks are cycled then why don't you just move them all over to the 120g ? Put the protein skimmer from your 60g to the 120g. If you don't have a protein skimmer at all, then just put an air stone in the tank to oxygenate the water. The fish will be OK without a protein skimmer so long as the water is oxygenated by other methods. There are people who run fish tanks successfully without protein skimmers believe it or not. You can also point a powerhead at the water surface to agitate it which will also help oxygen exchange.
One negative aspect with doing it in stages as you are suggesting is that you will likely stress out all the fish each time you do a move. If you are moving rock from the 60g to the 120g each time then I presume you are also aquascaping. This will stress everyone out as well.
At any rate, it certainly can be done in stages as you are attempting and if so, all your fish on that list are fairly hardy. The mandarin is probably the weakest link but only if he/she has trouble feeding.
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