Prior to 9-11, there were a few cases where passengers were allowed
international transit. After 9-11, this was stopped, and all passengers
are required to enter the USA in order to have the privilege to then
leave the USA.

I think there are still a couple of exceptions to this (namely Air
Canada YVR-HNL-SYD flight where they relaxed the ridiculous post 9-11
rules to only checking passports).

Since you are changing airlines and then boarding what is considered a
domestic flight for AA, you will definitely have to enter the USA
formally, and anyone will need proper papers to do that (visas etc).

Similarly, on the way back, you will be required to clear USA
customs/immigration at Vancouver prior to boarding a "domestic" AA
flight to Washington. Again, you will need proper papers to allow you
entry into the USA. (as you leave the AA flight, you are in a domestic
area of DFW and could leave the airport without any formalities).

Because both legs will be to the USA, the airlines will be forced to
send all your passenger's personal data to USA authorities, and they may
decide to prevent someone from boarding due to their name or whatever
other unpredictable reason, or worse, detain them upon arrival at DFW
and then send them to some 3rd world country for "rendition" (aka: Torture).