For experience based I haven't seen anything. It would be a nice thing to add, as that would make leveling way more interesting and I may even take note of it then.
This is the closest I have seen:

Event based case:
+minturn <turn nbr>
+chance <percent>
+season <0-3>
(0=summer, 1=autumn, 2=winter, 3=spring)
+seasondelay <1-3>
(3=early (guaranteed to only happen once per season), 2=mid, 1=late)
+varequal <var> <value>
(var = 0-999, true if variable equals value)
+varlesser <var> <value>
(true if the variable is less than value)
+minturn <turn nbr>


[Marlin]
Wow! It does seem that event triggers can be logically negated with a minus instead of a plus!
[carn112004]
E.g.:
squareevent
+terrain 127
-class -3 "Sylvan Queen"
alterterrain 4
endevent


or in defining a monster:
growtime <value>
(will grow into next monster after about value turns)
growoffs <value>
(to become other than next monster)

growoffs? I tried searching the docs, neither attribute is came up :/.

In the documentation post here, under:

Monster
-----------------

I wouldn't say I know they were implemented as expected but that said I imagine they've been tested.

There's no transform by experience, that is a dominions modding command.

As Zenithir mentioned "growtime" is best you are going to get in CoE.

Isn't there an evolve function that keys off of the units HP? Fiddling with that so that a unit that has gained enough EXP to gain HP will change into a different unit type should be possible.

I don't know is there? Would depend on which basis it worked off of. If it's final hp, it could get triggered by magic items.

I haven't looked at the actual, terribly arranged, code but the 40k Imperium of Man mod had
line troopers growing into special weapon toters. Magic items did screw with the evolution process. Back in COE3 the Draconian mod had dragon whelps that would swallow whole to inflate their hp values until they evolved.