Are you using IMAP or POP3 mailboxes?
When using IMAP, all your device(s) will download a copy of your email messages when fetched from your Mail providers email server. The Mail server will keep the original email messages until such time as they are proactively deleted from either your iPad or iPhone. When an email is read, it will show as such on all devices. IMAP is intended to keep mail and associated status consistent between all devices that can access the account.
If instead you are using POP3, email clients can be configured to (a) download and leave a copy on the server - or (b) download and delete from the server. As such, email messages may be deleted from the Mail Server as soon as they are downloaded by a single device. If you are using POP3 and downloaded mail is only saved on your old iPad, transferring archived email will require that you (a) create a full iCloud or iTunes backup of the old iPad - and (b) restore the backup to your new iPad.
How to back up your iPhone, iPad, and iPod touch - Apple Support
Restore all content to iPad from a backup - Apple Support