Why does my tube flash converter drift after warmup?

Your instinct is basicly right, but it's not the emission current itself so much as contact potential. In a 6AL5 the effective threshold where the diode starts conducting sits right around the cathode's contact potential, and that point moves as the cathode strip comes up to full operating temperture. The first half hour is exactly the window where the cathode sleeve, the mica spacers, and the glass envelope are all still soaking up heat and shifting the geometry and temperature of the emitting surface a few degrees at a time. Once everything reaches thermal equilibrium the contact potential stops crawling and your thresholds lock in. So yes, it's something tubes do, and it's worse in a comparator application because you're riding right on that knee where a few tenths of a volt of drift moves the code. The classic fix from the era is a heater standby arrangement. Leave the 6.3V heater winding on all the time, or switch it on with a seprate switch, and only close the B+ when you're ready to run the punch. A hot cathode with no plate voltage barely ages, and you'll find the tubes settle in maybe ten minutes on heaters alone instead of thirty with everything lit. Some old gear used a thermal delay tube or just a time delay relay on the B+ for exactly this reason.

A second trick that helps: run the heaters a touch low, like 5.8 to 6 volts, once they're warm. Lower cathode temperature means less drift for a given ambient change, at a small cost in emission headroom, which a comparator ladder doesn't need much of anyway.

Also check your reference ladder resistors. If those are carbon composition they drift with temperature too, and sitting right next to warm tubes they'll cook for the same half hour. Metal film or even good wirewound in the ladder, plus keeping the string physically away from the tube sockets, can cut the wander noticeably.

Honestly though, for a retro build feeding a paper tape punch, a heater standby switch is period-correct and solves it. Flip heaters on when you walk into the room, flip B+ when you sit down.

Agree with the standby switch route, that's what fixed it for me. I had the same creeping thresholds in a tube voltmeter I built years back and leaving the heaters on a separate toggle made a huge difference. Mine settled in about eight minutes on heaters alone, and I wired a little neon pilot lamp across the heater winding so I'd know at a glance the cathodes were cooked before I threw the high voltage on. Also found drift was way worse on cold mornings in my garage, so if your bench sits in a cool room expect the warmup to stretch longer until the sockets and chassis stop soaking heat.

This explains a lot actually. I have an old shortwave radio with the same lazy-in-the-morning behavior and I always figured the tubes were just bad. One thing I keep wondering though, if I leave those little heater wires powered up all day does that eat the tube life faster? My dad always said turning stuff on and off was what killed them, but I never knew if that was true or just something he said so we would stop fiddling with the dial.