What is the name of this type of directed graph problem?
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.