Ignoring your deliberately emotive hyperbole, yes. Subject to the immigration controls of any countries which they pass through they should be able to claim asylum in the country that they want to claim asylum in.
Fair enough, that's a nice direct answer
But I would disagree, on both ethical and practical grounds.
On ethical grounds: Basically, no-one else has the automatic right to decide which country they live in. If you (assuming you're a UK citizen with no other nationality) or I or almost any other UK citizen were to decide we fancy going to live in - say - the USA or Japan or China or almost anywhere other than a small number of countries, we would need to apply for a visa and convince that country that we have a reason to live there which meets their normal criteria for whom they are willing to accept. You seem to be saying that, if you become a refugee, that should therefore give you some kind of roaming rights that no-one else has. I simply don't see that. I believe all countries - including the UK - should do what they reasonably can to provide new homes for those who can no longer reasonably live in their home countries (because of war, persecution, etc.), but that's not the same as letting people effectively turn their noses up at perfectly safe countries because they'd rather live somewhere else.
And on practical grounds: The number of refugees plus the number of people across the World who would have legitimate claims for asylum if they put in a claim is staggeringly huge: Probably in the hundreds of millions, maybe over a billion. It is way, way, too much for any small group of countries to accept. If you adopt the principle that any refugee should be able to pick which country they want to live in, then the vast majority of refugees are (quite understandably) going to choose one of the few countries that has the highest standards of living, the highest quality of life, the most internationally recognised language, etc. etc. Having everyone choose a small number of countries is very obviously not workable.