As a former airline operations analyst with considerable involvement in schedules, I disagree that airline schedules are more complex than railroads (well, in some ways they are but in others they aren't). First, at least at my carrier, schedules are based in UTC (aka GMT) and are then converted to local. So the time change gets incorporated into the schedule as viewed in local time automatically.
Second, flights are managed by segment - a segment being a departure to an arrival. Even if the flight has multiple segments, each segment is managed on its own (if Amtrak ran a train by segment, a NYP-WAS Acela would be managed as NYP-NWK, then NWK-PHL, PHL-WIL, etc.). So an airline flight segment is affected by a time change only if it occurred during that segment; with Amtrak, a time change on the first night of a two-night train affects every segment downline of it until the train reaches it's terminus over 24 hours later.
Third, the only flights affected by time changes are red-eyes. At least for my carrier, that's a relatively small number domestically. And as for international, North America and Europe don't change on the same dates. While the U.S. moves off daylight time tonight, Europe did it a week ago so for my carrier, we had temporary schedules for all this week (in the Spring, Europe changes on the last Sunday of March). And then some countries don't change at all and it's largely backwards in the southern hemisphere.
So for tonight, let's say there is a San Francisco-Chicago flight that normally departs SFO at 11:30pm (0630 UTC) and arrives Chicago at 5:30am (1030 UTC). Tonight, it would still be in the schedule as 0630 to 1030 except the arrival time at Chicago would translate to 4:30am and that's what the passenger would see. And if the were then connecting to a flight further east at 7:00am, it's still going to depart at 7:00am so just like Amtrak, they get to sit for an hour only they do it at their connecting airport rather than having their flight hold for an hour.
Now if this were March when we turn the clocks ahead an hour, that normal 5:30am arrival becomes 6:30am. Does the passenger risk missing their connection? No, because the reservation system builds legal connections on the fly and never offers the customer the illegal 6:30am to 7:00am connection (or schedule planning sees the problem in advance and moves that 11:30pm departure up to 11:00pm for one night to restore the connections - and most people won't even realize it was changed for one night as I don't think any airline publishes full timetables anymore).