If an employee is scheduled for a mandatory training, and he or she is paid mileage to the airport (three hours away) and for the actual time spent in the training, are there also any requirements to pay travel time? If he or she is flying from MO to another state and has to spend hours on an airplane and a layover, should this person be compensated at least for time spent in the air?

Reimbursement for mileage (or not) is irrelevant to the compensable time issue.

If the employee is a passenger on an airplane, the time spent in the air is compensable time IF it occurs during the regularly scheduled hours for the employee, even if it occurs on a nonscheduled workday for that employee. The FLSA has not addressed the issue of travel time to the airport, being there early to go through security, etc. although, if it were me, I would pay them for it. After all, that is part of the travel, though the DOL is silent on the issue.
http://www.dol.gov/dol/allcfr/ESA/Title_29/Part_785/29CFR785.39.htm

Thanks, Pattymd.

If I am an emergency worker, and could be scheduled any time during the day, would then any time the travel time occurs count? Say from the minute the plane takes off in MO to the minute it lands at my destination?

Not under the FLSA. What type of "emergency worker" are you? And are you covered by a union contract?

No, I work for a city.

So, what type of emergency worker ARE you? (BTW, working for a city and being a union member are not mutually exclusive :) )

Sorry-- I'm a firefighter, no union contract.

My original answer stands then, at least as regards out-of-town for training.