I would tend to believe it is a good idea (to automatically run code formatters), but that is just my opinion.
I won't run them periodically, but if possible before version control commits.
With git, a pre-commit hook doing that would be useful. In many C or C++ projects built with some Makefile, I am adding some indent target (which run suitably code formatters like indent or astyle) and expect contributors to run make indent regularly. BTW, you can even add some make rules to ensure that the git hooks have been installed (or to install them).
But really, it is more a social issue than a technical one. You want your team to commit clean and nicely formatted code, and that is a social rule of your project. (There is not always a technical answer to every social problem).
Version control is mostly some tool to help communication between human developers (including your own self a few months from now). Your software don't need VC or formatting, but your team does.
BTW, different communities and different programming languages have different views on code formatting. For example, Go has only one code formatting style, but C or C++ have a lot of them.
