If you have an eg: IMAP, Exchange, Office 365 or iCloud account (non POP account), then you can access and see the same email on all your devices including phones and tablets etc, as all your messages (are stored on the mail server end) and not locally. So no messages are normally stored in eM Client for those type of accounts.
Yes you may need to buy mailbox cloud space if you have eg: too many messages that a free cloud mailbox space can handle. So all depends on how larger mailbox you have now or likely to get.
Most major providors though have inexpensive cloud mailbox space you can purchase, if you need more than a free cloud space. So you just need to shop around for what suits your needs best.