eM Client contains an Automatic Archiving feature which allows you to download all messages older than a selected number of days from the server and keep them stored on your local drive.

You can find this option under Menu > Tools > Settings > Mail > Automatic Archiving. Make sure to set Archiving scope under Account Preferences before you enable or perform archiving. With the archiving scope, you set what folders and accounts the archiving will be performed on. 

You can then set how often the archiving will be made and how old the messages affected must be. 

Note: The Archive button in the main toolbar is a separate feature created for Gmail accounts - it performs server-side archiving, ie. removing all labels and saving the message in All Mail folder.