Your turn is generally considered the amount of things you accomplish in a round, as in what you do in 6 seconds. So even though we break it out into serial order, (X happens, then Y happens) it's actually all happening at the same time(X happens while Y happens). Consider page 181 of the PHB
Every character and monster has a speed, which is
the distance in feet that the character or monster
can walk in 1 round.
You move that distance in a full round, not a fraction of a round. So a full 6 seconds, not in a fraction of 6 seconds. As such by the time he's finished moving and acting it's been 6 seconds and there is no more movement. He begins to fall next turn. This does seem to be at odds with 14th level eagle totem though
W hile raging, you have a flying speed equal to
your current walking speed. This benefit works only in
short bursts; you fall if you end your turn in the air and
nothing else is holding you aloft.
This seems to be indicating that you should fall that distance this round, otherwise you just be able to start flying again next round and never hit the ground(which I assume is what he's trying to do). Personally I'd ban that as violating RaI, since it's clearly meant to indicate that this ability is not true flight. If not banned it would seem you have to come up with some ruling that indicates falling and therefore ignore the assumption that a turn takes the entirety of a round.
In that mode I would consider that movement is probably 2 seconds. On your turn you can perform a move and an action. An action is likely equal to a move in time because Dash lets you move again, indicating they are similar in time. You also have a reaction, the ability to interact with an object, and perhaps a bonus action. Those all are less involved things than an action or movement and thus I'd say they add up to 2 seconds.
While the text specifies end of turn the fluff really seems to suggest end of movement. You get a burst of flight, not the ability to hover. So while the abstraction will allow you to fly, attack, and fly again with a RaW interpretation, we can still assume that you are falling for all the time not spent actively flying. So 4 seconds, unless the player takes dash, in which case 2. If a character falls 4 seconds they will fall 257 feet. In 2 seconds they'll fall 64 feetcalculator, assuming earth gravity. So if they Dash up to 80 feet they should be safe, provided this breakdown of a round.
Alternative breakdown of a round is that each turn takes the same amount of time and they do happen all in serial. In which case, no matter what he'll hit the ground if he does this as the max amount of time his turn could take is 3 seconds in a 2 person encounter leaving a minimum of 144 feet of falling distance.