Do they get to pick and choose where to flee?

This is way too broad to be useful

They do have a right to "seek" they don't have a right to live in the country in question.

The critical point is that people are traversing several countries or seas to apply for asylum in the EU. This is obviously not following the spirit of the applicable laws which were designed for avoiding humanitarian tragedies in neighboring countries.

Many aren't even persecuted, just poor, lacking education and desperate.