Many longitudinal models with mediators do not specifically test whether a change in X leads to a change in M which then leads to a change in Y (or for experimental setups whether getting the treatment leads to changes in M which leads to changes in Y).
The only tool that specifically does this is the one using the Latent Change Scores, here are the first published such examples (parallel LGM are doing something like this, but commonly linking synchronous changes). One can also take a look at http://www.modeling.uconn.edu/archive/20131/2013/ under Testing
Mediation the Way it Was Meant to be: Changes Leading to Changes then
to Other Changes. Dynamic Mediation Implemented with Latent Change
Here are the first 2 examples visually:
which should have then specified the green paths I added: