What clustering algorithm is appropriate for clustering paths?
Like Ricardo mentioned in his comment on your question, the main step here is finding a distance metric between paths. Then you can experiment with different clustering algorithms and see what works. 

What comes to mind is dynamic time warping (DTW). DTW gives you a way to find a measure of "distance" (it is actually not strictly a distance metric, but it is close) between two time series. One very useful thing is that it can be used to compare two time series that are of different lengths. 

There are many good blog posts on DTW, so I won't try to give yet another explanation of it. There are also many python implementations of it. And a lot of work has gone into making the algorithm very fast. DTW is a strange algorithm--in some ways very simplistic, but typically works well. Once you modify the algorithm to deal with paths, you can construct the distance matrix and use that for clustering. One common clustering algorithm that is used in conjunction with DTW is spectral clustering, since the distance matrix can be used directly (instead of the matrix of data points, which we don't have here).