Does the onus for dealing with these migrants and refugees lie in the home country of the refugees (such as syria or sudan), the transitionary countries (such as turkey or libya), or with the destination countries (such as italy or germany, or the european union as a whole)? why?

The onus of dealing with the refugees to begin with can not be placed on the originating countries. This is because these countries are in civil war to begin with.The transitional countries can not help either,because certainly most of these have considerable levels of failures as nations.
the onus falls on western countries, because, that is where the refugees are going. It is upon these countries to come up with strategy of coping up with the refugee rides across the Mediterranean.

The best way to deal with the crisis is to put up transitional camps in the countries where the refugees are emanating from.

Transcontinental railroad was the most effective way to travel.