The Dublin regulation in the EU specifies "first country". There's the idea of a "safe 3rd country" lurking in many publications, as well. Notice that Canada's picked up the EU's Dublin regulation. It's unclear that refugees have all the rights and the country they're aiming for has none.
The 1951 Convention says that you can't kick a refugee out--an "economic refugee", by the way, is a "migrant" and not a "refugee" at all for the purposes of that convention--even if s/he entered illegally. It doesn't say you have to admit them. That's seeking emanations from penumbras that are implied by what's left out of the text. Its text allows for "first safe country" and "third safe country" implementations.
However, it says, article 31,
1. The Contracting States shall not impose penalties, on account of their
illegal entry or presence, on refugees who, coming directly from a territory
where their life or freedom was threatened in the sense of article 1, enter or
are present in their territory without authorization, provided they present
themselves without delay to the authorities and show good cause for their
illegal entry or presence.
2. The Contracting States shall not apply to the movements of such refugees
restrictions other than those which are necessary and such restrictions
shall only be applied until their status in the country is regularized or they
obtain admission into another country. The Contracting States shall allow
such refugees a reasonable period and all the necessary facilities to obtain
admission into another country.
Notice the word "directly." "Directly" is like the term "persecution". It's not defined in the text. How you define it depends on whether you place first the desires of the alleged refugees or the purpose of the treaty--safety for the refugees.
Treaties like this are bad laws for implementation. They're mostly to guide national laws, and govern less interactions between people at the border and more the actions of legislatures.
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