Transition plot in R how does it work?
The problem with Sankey diagram in your case is that you will duplicate all the nodes (on the left, and on the right), that is messy.

Networks are better, but nodes are not well structured.

I suggest to try chord diagrams for such data. 



One of the greatest examples of chord diagram was build with d3.js, and based on migration flows dataset. Here is an R wrapper for d3.js chords. Or you can try to reproduce this manual.