If they stay then odds are in your favor. I have, though, had more than one queenless swarm stay put and try to set up housekeeping. The other bad thing to watch for is they may try to bail on you if for whatever reason they don't like the box you put them in. You will usually know in the first 24 hours if that going to happen. If your lucky they will just all spill out and lay in a wad next to the box if not they go full swarm and head for the trees. If you catch them trying that stunt just catch them again and throw them back in but this time set your queen excluder under the brood box to hem up her majesty.
I, too, would feed at this time. They are burning carbs to make wax and will quickly burn through their on board stores. By feeding you are saving them flight time to gather nectar and comb gets pulled faster and mama has more places to lay in a shorter time.
This is what you should expect. Day 1, very little activity at the entrance, maybe a guard or three with a few exploratory flights. Just check enough to be sure the bees are in there. Day 2 will see a little more activity and by days end you might even see some pollen coming in. Day 3 should see regular flights getting established and pollen. Of course all this is contingent on the weather. The main thing I look for is pollen as they do not bring pollen when they swarm, it must be gathered at their new site. If I got pollen I'm comfortable calling them definitely caught.
Now let them be until the end of the week, other than topping off the Boardman feeder as needed, at which time you'll take a quick peek inside. Slide everything to one side then go straight to the frame at the very center of the cluster, push the neighboring frames away and withdraw the center frame. The it should be welled pulled over most if not all of it. You need to see eggs down in the bottoms of those cells. Don't worry about the queen, if you have eggs you have a queen. Slide the frame back in and shove everything back to the center and close her up.
You'll check again in a week's time and begin your manipulations then such as rotating frames to the center, checking brood patterns and the like.
Hope this helps.