Mac OS X Mail included with EVERY version of Mac OS X has always allowed the desktop presentation of Mail from your ISP and other Mail services, even across multiple accounts. Browser-Mail /Web-Mail is not required.
If you upgrade Mac OS X, you will get a different version of Mail, which provides a built-in one-way conversion of your older Mail messages and accounts to newer mail formats, but will not allow the older mail formats to continue to be used.
Newer versions of Mail will not run in older versions of Mac OS X.
Older versions of Mail will not run in newer versions of Mac OS X.
You have not been particualrly clear about exactly what you are trying to accomplish.
Try describing what you want as an outcome as if explaining to a child, or ask a much more focused question.