We launched a mobile site a couples months ago following the parallel mobile structure with a URL:m.example.com

The week later my crawl detected thousands of dups which I resolved by implementing canonical tags on the mobile version and rel=alternate onto the desktop version. The problem here is that I still also got Dups from that got generated by the CMS.

?device=mobile

?device=desktop

One of the options to resolve those is to add canonicals on the desktop versions as well on top of the rel=alternate tag we just implemented. So my question here:

is it dangerous to add rel=canonical and rel=alternate tags on the desktop version of the site or not? will it disrupt the rel=canonical on mobile?

Thanks