Recently, I found the following presentation by Darij Grinberg about applications of diamond lemma. Here is the link:
http://www.cip.ifi.lmu.de/~grinberg/algebra/diamond-talk.pdf
It seems to be very interesting tool for combinatorial problems (which deals with some process). I want to learn more about it and I am interested in examples of combinatorial applications of the diamond lemma. I also find some examples (including algebraic examples) here
https://sbseminar.wordpress.com/2009/11/20/the-diamond-lemma/