It may surprise you to know that in at least one EU member state, this is already the case.
I recently found myself sans employment here in Germany and found that you simply MUST register with the local equivalent of the job centre as 'Job seeking' as sson as you are unemployed. They REALLY like to know peoples business here. If you don't then you have to start paying your own medical insurance at full price. I found that out the hard way.
So anyway, I did the registration thing. Only to be told "you haven't had 12 months employment here in Germany, so we must request accounts for your own government and see if they will cover the costs of your unemployment."
Follows a long and entirely unpaid wait for the German bureaucrats to talk with the British ones to find my status. Long story short, the UK has to pay any benefits I claim.
I just found a crappy job to fill the gap. Things happen at the speed of mould growth in the desert here.