According to Southwest's policy, people travelling together but with different boarding positions have the option to board together, provided the person higher up in line waits with the people further back. How this applies to families, I'm fuzzy on, but I would assume if you have a business select or other pass that allows boarding in the A1 through A15 group, it would make sense to have young children (say, under 10 years old) board with you. It seems like this is what the guy had done on several flights previously.
I have tried to find this regulation, recently, but I am fairly certain that that TSA regulations require the airline to honor any preferences and boarding assignments for all minors with the ticketed adult they traveling with, or else treat the minors as unaccompanied minors. A few times I have flow and received an automatic upgrade to first class, but not my minor travel companion. It's actually turned out to be a hassle, because they filled my old seat, there by stranding my child next to a random stranger. Obviously not acceptable. In both cases that it's happened, they ended up trading the stand-by person into my first class seat and putting me back into coach (which was fine). However, in both cases they told me that TSA regulation required them to board us together, and to use the adult traveler's cabin class, boarding priority, and seating preference. I travel for business quite a bit and I wouldn't wish my kids into 1st class with others who are traveling for business so I was happy to move back to coach, but both times the gate agent was quite adamant that they would bump a 1st class passenger to follow the reg if I wanted.
I sort of think this can't true. Someone from the TSA told me it had to with a terrorist screening database, that the children were not entered anyways, so it worked around a problem in the database with moving passengers between cabins. Could just be a myth.