It depends on the email system that an email account uses. The original email system is Post Office Protocol (POP); this is largely being replaced by Internet Message Access Protocol (IMAP). The main difference between the two is that POP downloads messages from the mail server and stores them locally (eg. In the Mail application of an iMac or Macbook), while, if your email account is IMAP, you simply view messages that are on the mail server. The messages remain on the server so that other devices that you might have can also view them.
In practice, IMAP messages (read, sent or deleted) can be synchronised across different devices whereas such synchronisation is usually not possible with POP email accounts, even though some POP systems allow you to keep a message on the server after it has been downloaded. So, if an IMAP account has been set up on all your devices, they should all show the same messages after they synchronise. However, this will not be the case with POP accounts.
I have found that some of my email accounts get set as POP on one device and IMAP on others so there isn't complete synchronisation. My guess is that your accounts are not all set to IMAP.
By the way, there is a third system used by Outlook.com accounts. This is called "Exchange" and will allow synchronisation.