I need help with moving a character in Unity2D c# but just along a path

So i'm working on a project where the player drags and drops an object on a path and the player character has to move to the dropped object. The problem is that the character object should not try to take the shortest route possible but stick to one particular path. I tried using nodes and pathfinder but the character stops at each node along the path. The game is based around isometric 2D puzzles where the player has to guide a character to an exit by dropping objects to eat.

I would really appreciate some help as I've been stuck on this for more than a month now without success.