Create a new folder in Local Folders/On My Mac. Name it whatever you'd like. Select and drag each message or a selection of message to this new folder. Once copied, you can delete those messages from the mail server to free up space.
Select an existing folder or put the messages you want to move into a folder in your mail account on the server.
Then, when done, drag that folder to Local Folders/On My Mac.
Watch the status messages so you know when the copy is complete. Large folders might take a while. (Apple Mail users, select Window -> Activity to show the activity monitor).
Once the folder has been completely copied , you can delete the folder from the mail server (right click the folder and select 'delete'). If this folder doesn't change (i.e. - it's an archive of old mail), you can copy it to other computers as well, if you have more than one so you can see it there as well.
You can still search this mail, but now you do so from the local folder, instead of on the mail server.