What GMorrison has included is helpful I'm sure but it only tells part of the story. On the attack transports and probably others, the Welin davits were "triple banked" meaning that one set of davits was used to stow three boats. One one deck, one above it on swing out cradles, and the third hung over the side on the falls.
So from the point described in the text the procedure would be:
Pins in place the falls were then reversed to let the first boat down to rest in cradles on the deck. The falls were then raised to their top position. The pins were now removed and the arms travelled outboard. When they reached their outboard position the falls would be dropped to pick up the second boat.
As done for the first boat the second boat would be raised and brought inboard. Once the davit arms reached the top position the pins would again be inserted. At this point a second set of cradles would be swung into postion over the first boat and under the second one. Pins out the second boat would be lowered into the upper cradles and the falls released.
The falls would be fully raised and the pins removed. The davit arms were now free to travel to their outboard position where the falls would now drop to pick up the third boat. The third boat would be raised to its upper position but it would be left hanging from the falls outboard of the ship. It would be gripped in and left in this postion as a ready lifeboat.
Sounds more complicated than it really was. When the command came "away all boats" all three boats in the triple bank cold be launched in about ten minutes.
There is more to it. The davit arms were connected by a horizontal beam. If I remember correctly the lifting falls were hung from pullies in the beam although the rigging went back through the beam to the davit arms. There also was a heavy weather pendant that could be used to pick up a boat above high waves although I don't remember how it was then transferred to the normal lifting tackle.