I’m not familiar with that setup, but most email programs have an option to export the data to some kind of text file format that can then be imported into another program. The downside is that you lose all formatting and attachments.
You may be able to export to a format that is compatible with your new system and saves all the data intact, especially if you are going Microsoft to Microsoft.
It all depends on the source software (email client) and the destination software.
Without more detail I can’t tell you much more than that.
going from outlook express(vista-old laptop) to Thunderbird(8.1-new laptop)