You will find those who love or hate each of those, with voluminous reasons for doing so. But they are all fine engines if reasonably cared for, and I wouldn't turn down a truck I liked because it had engine rather than another.
Cummins of course is a highly respected name in diesels, but the Ford is a Navistar engine with Ford plumbing and the Duromax is an Izuzu engine, another company that is well versed in small diesels. The 6.0L Ford is the newest design of the three, so it might have a few roiugh edges yet, but I haven't heard anything like a serious problem with them.
You should also consider the tranmission tied to the engine. If you are going with an automatic, the Duromax+Allison combo is hard to beat. Ford has a nice new 5 speed auto now and it appears to be a good one, but again it has only been out a couple years. The Dodge trucks used to have the weakest automatic of the three, a problem Chrysler has been striving to fix. I don't know how much progress they have made so far.
Roamer is correct, you can't really go wrong with any of them. Engine technology is progressing as fast as computer technology, well almost. I found the best way to find out which has the least problems is to make friends with a tow truck driver. Ask which gets towed the most. For a while it was Dodge than Ford and the least was Chevy, but that was before the Dynamax engine came out and Chevy was in the running for the tow vehicles. JMTCW
One thing I learned from my time as Shop Foreman for a Big Three dealer:
They all break.
I was a dyed in the wool GM man forever. Then came their 350D fiasco. I love the look of the Ford Super-Duty. Ford is working with Cummins for their next generation consumer diesel engine. Had a bunch of friends who owned Chrysler products with automatic transmissions and they were on a first name basis with their tow truck dispatcher.
Bottom line is that we all decide on what's "best" for our situation. The race for torque and horsepower changes leaders every year but all of the diesels produce half again more than the best you could get 5 years ago. My advice is to decide what's most important to you and get your butt in some seats. You'll quickly develop a preference based on your test drives.
<I am looking to purchase a new Ford, GMC, or Chevrolet pickup truck to move a 5th wheel of about 15,000 lbs.>
You may trying to bite off more than the truck can chew. You did not specify the truck size but if it's 3/4 or 1 ton I suspect the pin weight will overload the truck, but the engine will pull it not stop it.
I agree with most of the comments about the trucks, especially about a 15,000 lb trailer being pretty big for even a 1 ton. If you add the approx. 7500 lb weight of a one ton dually with 4WD to your trailer, you are at or above the limits of all the trucks I know about. At up to 25% of the trailer weight on the pin, you are looking at 3750 lb which will pretty much max out any 1T.
Also, IMHO if you are going to tow, the manual transmission is the way to go. I know that the Allison is an awesome auto tranny, but I still prefer the control I have with my manual six speed (Dodge 1T DWD 4X4 CTD).