Invalid key error when browsing through my sites

Hello everyone!

I have an issue that is very annoying for me as the administrator of my many sites.
When I login into one of them simply going to /wp-admin/ everything works fine. When I'm still logged in as the admin and I go to another site I see the site for a second, than the screen goes to Wordpress Admin Design and gives me a message:

Unknown login key

The URL becomes very long and lookes like this:

http://www.company.com/?action=login&dm=BIG_NUMBER_HERE&k=BIG_NUMBER_HERE&t=BIG_NUMBER_HERE

I would really like to fix this issue.
I'm attaching a screenshot of how this all looks.

The strange issue is that when I change the long URL to just /wp-admin/ it simply works.