There is more to it than just the Euro. The Euro inevitably plays a part, but there are countries not in the Euro who have played a huge part in the migration across Europe.
You really have to look at push and pull factors.
The Euro has created an environment that has enhanced the pull factors for example in Germany and the push factors in Italy but those factors are just the same in non-Euro countries.
Those push and pull factors elsewhere are managed by systems such as the ones operated by Australia.
They are seen within the UK, even within regions of the UK.