Apple iPhone or iPad

You can shorten the number of days your email app syncs mail for:

The setting is specific to each email account on your phone.

MS Exchange and some IMAP servers support this feature. For POP, see further down

MS Exchange and some IMAP servers

• Go to Settings/Mail/Accounts and tap on your email account.
  If the account supports it you will see an entry for “mail days to sync”.
• Shorten it to a week or two.

If you don’t see it the account will always sync all mail. Just be patient.

if you are still using a POP account

• Go to Settings > Apps > Mail > Accounts
• Select your POP3 account
• Tap Advanced to find the "Delete from server" setting.

if it is set to Never, change it to a shorter time, like 7 days.

Note for a more reliable and synchronized experience across multiple devices, it is generally recommended to use the IMAP protocol, which is designed to keep emails on the server by default.