It depends on the pen, and it depends on whether there are any changes
in cabin air pressure during the flight. I've had piston fillers leak
on one flight and not on the next (same Parker Sonnet both times), I've
had pens with cartridges fitted that were ok one flight and not the
next (specifically a Parker 45 and a Lamy Safari). The pens I would
trust the least are aerometric fillers and lever fills (Hero pens from
China, in my case). The only pen that has never leaked during flight is
my Parker Duofold. Possibly that's because the Duofold's cap screws on
and might maintain the air pressure inside the pen, or as least slow
down the rate of pressure change so that things equalize more gently
and don't cause the ink to bleed out through the nib. All the other
pens have click-on or slide-on caps.
I've never had a pen that was damaged by changes of air pressure,
although I guess that's possible. They have just leaked or dropped big
blobs of ink, especially when used in flight (the Parker 45 leaked into
the cap -- ugly surprise when I opened it afterward). Now when I carry
a fountain pen on a plane, it will either be completely empty
(preferred) or completely full, and I never use it in flight. I also
never carry the pen in my shirt pocket or pants pocket (even if in a
pen case). I have a small leather pouch fitted with an ink-absorbing
lining and internal sleeves for four pens (or Visconti travelling ink
pots, which fit perfectly) and a pocket for refills; the pens and ink
pots stay in the pouch during flight. I am very careful when I open the
pens after flight, and only do so over a sink (not in a client meeting,
which is where I opened that Parker 45, unfortunately). The Visconti
ink pots have never given me any trouble, leakage-wise, and they allow
me to empty any aerometric, lever-fill, or converter-equipped pen
without wasting ink.
If you fly a lot and want a pen just for travel, you could always get
one of those eyedropper pens with a safety shutoff valve built in. As
long as you remember to close the shutoff valve before you fly, you
ought to be ok. They hold a lot of ink, too (could leave the ink pot
home).
Hope this helps.
Bill