Hi ThirdCultureKid,
There are a couple of important factors related to email that must be known to understand or isolate the behavior you're describing:
1) Your email service provider is where your email messages arrive or are sent to others.
2) Your devices are set up with the address and credentials necessary to check the service provider account for downloading or uploading emails.
3) Your email provider may allow POP access to the server or IMAP access, or perhaps allow you to choose which method to use.
4) POP email access allows each device to download the messages from the server and upload outgoing emails one time. IMAP access works to synchronize mailbox and message changes between your device and the server on a continual basis.
Knowing this, if your email account is set up for POP access, the behavior you describe is expected and normal. Each device will log into the email server, download new messages, and log out. On the next connection, they will download any new messages. IMAP access would be necessary to have changes made on one device reflect on others.
If you're unsure how your accounts are configured, use the steps from View email account information in Mail on Mac to view that on a Mac, or contact your email provider and ask if they allow only one type or the other.
Regards.