Getting a model to follow a path

A rigged model can easily walk along a straight path left to right.
Nothing new here.
But I next created a path (Bezier curve) and tried two ways to get the model to follow the new path:

  1. Parented the rig (treated as a single object) to the curve, choosing the option ‘follow path’.
    But with Alt+A the model continues withe its old straight path.
    On the other hand, if a cube is parented to the curve, it nicely follows the curve (without any key frames).
    Can parenting work with a rigged model?

  2. Instead of parenting, I selected the rig and applied the ‘Follow path’ constraint. This can be used both as an Object or bone constraint. The target here is the curve; I ticked the ‘follow curve’ option and chose some combination of Y, Z buttons (for ofrward/Up).

But the model refuses to follow the path and insists on its original straight path.

I may be on the wrong track here and would be most grateful for the proper way to force a model to follow a specified path after it has been key framed to move straight.
See screen shot .