Irrespective of whether the emails are kept on the server or removed after downloading locally depends on the configuration of the incoming server of your email client.
For the IMAP email accounts – All the messages will be retained on the server until they are deleted.
If you are using email hosting through POP3, you get the option to select if the emails should be retained on the server or they should be downloaded on the computer where the email client is working and then removed from the server. The exact location to change this configuration varies on the basis of the email client you are using.
Here are some examples:
For Thunderbird, while using POP3 account – go to the Tools > Account Settings > Server Settings for the account > and click on ‘Leave messages on the server’ option.
For Outlook – Under the option of internet email settings > go to Advanced Tab > and click on ‘Leave a copy of the messages on the server’ option.
When those options are checked, the email client will download the messages locally and also keep them on the server for a specified period of time until they are manually deleted.
This is illustrated in 2-sub options mentioned below for both the clients:
If the ‘Leave messages on the server’ option is not selected, the messages will be automatically deleted from the server when they are downloaded locally by the email client.