To legally claim asylum in a schengen state which has no external border to non-schengen states you literally have to fall out of the sky or travel into the country on a tourist visa.
Guess what happens in the moment a crisis breaks out like it happened in Syria? Yes right, all the wealthy northern european states do not give out tourist visas anymore.
Again, seeking asylum and immigration are two very different processes. When seeking asylum, you don't get to pick the countries; first safe is it.
They can however apply for refugee status in the closest refugee camp outside of Syria, usually in Turkey.
Sure you can go to turkey - where you have no chance to legalize your status in mid- or long term. Where your children are not allowed to go to school.
A constitution doesn't allow for blanket emigration.
There is always paper work involved to justify and motivate the request as well as quotas.
Countries can't afford to house and feed limitless amounts off refugees.