I think there are multiple ways to do what you want. For me, the most obvious one is storing tests for different browsers in different directories, and then just starting TestCafe separately for each directory. If you want to achieve it in a one TestCafe session, you can retrieve user agents in tests and skip tests that shouldn't be run for a specific useragent: forum question.
If you don't have to run all the tests in a one TestCafe session, you can install a copy of TestCafe on each machine, use the report file feature, and then upload report files to your main machine using FTP, SSH, etc. Otherwise you have to start your browser manually and use the remote browsers feature, or automate browser starting process by implementing your own browser provider.