Deleted e-mail

[#1]

I use Apple's Mail to access all e-mail accounts.  One reason for moving to Plusnet was because I had understood that, if I set up my Plusnet account with imap, deleted or e-mails moved from my Inbox would be removed from the Inbox of all my devices using that e-mail account.  This does no seem to be the case.
There are a number of possible solutions, the most drastic being to remove my Plusnet account from all but one of my devices.  As an Apple user I already have an iCloud account which removes e-mails from the inboxes of all devices but I tend to use that for family, friends and people with whom I want to be in regular contact.  I would prefer not to be bombarded with less important e-mails on the move.
Have I misunderstood how imap e-mail works?

[#2]

Quote from: colinparker

....One reason for moving to Plusnet was because I had understood that, if I set up my Plusnet account with imap, deleted or e-mails moved from my Inbox would be removed from the Inbox of all my devices using that e-mail account.  This does no seem to be the case.

Near....  IMAP access fetches the e-mails from the server as needed and should not keep them on the accessing device (similar to webmail in that respect), so deleting an e-mail removes it from the server. If devices are retaining the e-mail, then the software is making a copy, which is not a Plusnet problem but a software configuration problem (I get the impression, unverified, that mobile phones keep a copy and delete this when synchronising).
E-mails that are moved (including moved to trash) are deleted from the server inbox and written to the new folder, which may be on the server or local.
Phil

[#3]

It may be necessary to compact folders to stop deleted emails being displayed. Deleted emails are just marked as such by IMAP but physically remain in place until the space is reused.