How do I keep the development MySQL schema in synch with the live one?

This for me sounds that you need to establish a ground source of truth, not only for changes to the DB schema but for all the other things too. Ideally you can build up a new environment from all the ground sources of truth and it will match the current state of the production environment.

For an SQL schema, I would propose that all modifications to the schema go through SQL text files which are kept under source control and applied one by one. For example App::sqitch purports to do that, but you can also do it manually or with a script of your own.

If you can keep to moving all changes to your production environment through text files source control, you can basically replay and replicate the environment everywhere where you can replicate your source control to.