> [...] if these people are genuinely in fear of their lives, if they're running away from somewhere, why are they not claiming asylum in the first peaceful, developed country they arrive at?
Sorry but that's totally unrealistic. You're making assumptions that do not hold in real life. A refugee which will seek asylum from Greek/Italian authorities will end-up 99.99% in a concentration camp. He might very well be beaten first. Police brutality in Greece is renowned.
> There are slum-like camps in some parts of northern France that are full of people trying to find a way to smuggle themselves over the sea to the UK. This makes little sense to me, unless they are really economic migrants, not refugees at all.
They just try to go, to the place which they believe will be better for them. They might have a family, or some sort of link there, or they just heard from someone that it's better in London.
> Perhaps we should have some sort of europe-wide policy, where people must declare as soon as they arrive in the EU, and then each country agrees to take a quota so that the southern/eastern countries aren't swamped.
Perhaps it should be, or (given the raise of right-wing anti-EU all over Europe[1]) maybe it's just too late. The EU was marketed as something, but clearly turned out to be something else.
[1] Marrie-Lepen in France (clearly anti-EU), Syriza (is pro-EU officially, but there is a strong left-wing anti-EU segment inside the party) in Greece, Grillo in Italy (clearly anti-EU), UKIP in the UK (clearly anti-EU), not sure about 'Podemos' policy towards EU in Spain but they can't be very positive.