The inbox is full delete the mails Keep mailboxes from getting too big in Mail on Mac - Apple Support
How to transfer mails from Mac to iPhone / iPad , even they are signed in with same Apple ID and password say example1@icloud.com
In iPhone go to settings > in password and accounts > click on add account you can select the desired account it could be gmail , enter gmail address and password and sign in it will prompt to check calendar , mail , notes , slide the box of it the apps data will start syncing .
There is an option fetch new data - push > in iPhone
You can connect the iPhone and Mac using usb cable and sync them Sync content between your Mac and iPhone or iPad over Wi-Fi - Apple Support
Sync content between your Mac and iPhone or iPad - Apple Support (IN)
The mails from Mac / iPhone will sync and take time ( the fastest way is sync complete data ) they will be mirrored , you delete a mail from Mac it deletes in iPhone , make sure all the mails should be read ( no blue dots on each mail message ) Mark emails as read or unread in Mail on Mac - Apple Support
See the articles to delete mails from Mac Delete emails in Mail on Mac - Apple Support
And from iPhone Delete emails on your iPhone or iPad - Apple Support