If a "refugee" has fled to a safe country (Greece, France, Italy, etc.) and then tries to carry on to another country, they are to all intents & purposes personally redefining their status to that of an economic migrant, and should be treated as such.
However, the EU has a wonderful "open-door" law which states that "asylum seekers have no legal duty to claim asylum in the first EU state they reach."
It's no wonder that economic migrants from Afghanistan, Iran, and Iraq are journeying through numerous safe countries in order to reach the English channel.
The threat of automatic deportation back to their country of origin would be a good incentive for them to settle in the country which initially grants them the "asylum" which they allegedly need.
Greece, Italy, & Malta would then try and force the EU to review its refugee policies.