Assigning to Categories and Favourites - issues

When I assign emails to a category, e.g. Updates, this does not apply consistently to the mails if viewed on mobile.
Likewise, if I add a folder to Favourites on Desktop it does not appear in Favourites on mobile.
I would expect the assignment of categories and chosen Favourites to be consistent irrespective of the device on which they are being viewed.
If these reasonable expectations aren’t possible currently are they on the roadmap to be sorted soon?

These things don’t sync back to the server, so you won’t see them on other devices connecting to the same account.

Except Categories for Gmail and Google Workspace accounts, which are determined by the server so they should be the same on all devices.

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Thanks for the response.
My current understanding is that when I assign a category a rule is created for future mails. So am I wrong to expect those rules (and any other rules I create on Desktop) to be consistently applied to a message irrespective of the device in use?