What is the name of this type of directed graph problem? Take a directed graph $G$ where the edges are decorated with a a natural  number. We want the set of all paths $P$ between two vertices $v_1$ and $v_2$ such that each successive edge in the path is decorated with a natural number that is greater than the natural number decorating the previous edge. 

An application for this would be bus or train schedules. If you're trying to determine the different routes between two cities based on transfers between stations. (You can't take a second train scheduled to leave for before the first one arrives.)

I've informally been calling this a "scheduled graph". But I don't know what the name for this in the literature is.

Any references to algorithms related to this are of interest as well.
As far as I know, the problem is sometimes called "nondecreasing paths", and was studied since the 50s.
See for instance this paper: G. J. Minty. A Variant on the Shortest-Route Problem, Operations Research, 6(6):882–883, 1958.

The version of the problem which asks for the nondecreasing path from $s$ to $t$ with minimum last edge weight is typically referred to as "earliest arrivals" (since the last edge weight is the arrival time in the itinerary application). There's a lot of work in the literature focusing on this version.