Shortest distance problem with length as functions of time
This is known as the "time-dependent shortest path" problem.  Indeed research has been done for this problem; see for example the classical paper by Orda and Rom, and this recent SODA paper which proves that when the weight function of each edge is polynomial-size piecewise-linear, then the shortest path between two fixed points changes $n^{\Theta(\log n)}$ times.

Dijkstra's algorithm can indeed be used for this problem, when the waiting policy is imposed, that is, wait at a node if that reduces the final arriving time.  Without the waiting policy the situation is much wilder: the shortest path may not be simple, subpath of a shortest path might not be the shortest between the two endpoints of the subpath, paths traversing through infinite number of edges may have finite arriving time, etc.  See again the paper by Orda and Rom for more discussion.