Why is the fluorescence spectrum of pyrene time-dependent?
I think that you have rediscovered the emission of pyrene exciplexes :)

As you have correctly assigned, the structured fluorescence spectrum shortly after the pulse is the emission of pyrene monomers.

Given sufficient time (100 ns), an excited molecule of pyrene diffuses towards a pyrene molecule in its ground state and forms an exciplex. 

Fluorescence emission from these exciplexes is known to be broad and unstructured.