Article 14 of the Universal Declaration of Human Rights states that "Everyone has the right to seek and to enjoy in other countries asylum from persecution."
Do they get to pick and choose where to flee?
This is way too broad to be useful
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.
They do have a right to "seek" they don't have a right to live in the country in question.