Is there a generic word in English that means "through time"?

Solution 1:

'Diachronically' would stress that the action occurs through time:

"[f. Gr. διά throughout, during + χρόν-ος time + -ic.]

1.1 Lasting through time, or during the existing period." -- OED

Solution 2:

There's a veritable smorgasbord of words in English describing something that spans a lot of time. Amongst them are: lengthy, long-lasting, drawn-out, long-winded, overlong, and enduring.

If you want a word that simply refers to the fact that something happens over a period of time rather than forever, then temporal, as you mentioned, seems to me to be the word to use. Here's a dictionary definition:

3. enduring for a time only; temporary; transitory ( opposed to eternal ).

Example usage:

It changes temporally.

As the definition says, there's also temporary and transitory.

I also just thought of transient, though that points instead to something lasting a short period of time.

Solution 3:

I see that nobody has posted the most obvious (to me) alternative to temporal: chronological.

2. In units of time. [...]

Usage notes
In the "units of time" sense, the term is almost exclusively used to clarify a contrast.

Solution 4:

Perhaps use duration or durational.

Solution 5:

I'm not sure what you're looking for. Would indefinitely work?