So it doesn't spill over and hijack other threads...
Apskip may have some of this already. But I assume the first leg will be a flight into Buenos Aires, then moving on from there towards another place in Argentina? If so, the two most likely flights from Los Angeles, leaving half an hour apart and arriving an hour apart, are:
AA 276 from Los Angeles to Miami, 2255-0640 (confirmed)
AA 961 from Miami to Buenos Aires, 1055-2150
(There is AR 1303 from Miami to Buenos Aires EZE, 0900-1950, but it's a different airline with no code-share with AA.)
AM 469 from Los Angeles to Mexico City, 2325-0455
AM 30 from Mexico City to Buenos Aires EZE, 0830-2050
Also possible is a flight via Panama City (CM 303 to CM 279 or CM 363), but the first connection's transfer time is too short (38min) and the second one arrives a little too late (0056 the next day).
LP 2605 flies from Los Angeles to Lima to Buenos Aires, but it arrives at 1515, too early for us.
Other flight connections are possible, but they either require changes to different airline alliances or leave/arrive at strange times.
The AV 88 that Delgado was on left from EZE and not AEP, so teams must have been on this:
PZ 704 from Buenos Aires EZE to Asuncion, 0845-1035
It'd be really strange to have a team check in so late at night, could someone have been caught on a later flight?
AR 1214/PU 703 from Buenos Aires EZE via Montevideo to Asuncion, 0940-1435
PU 366/PU 703 from Buenos Aires EZE via Montevideo to Asuncion, 1040-1435
G3 7431 from Buenos Aires EZE to Asuncion, 1435-1625