And again (not to ghostys) to reiterate the point that illegal immigrants get NOTHING. Not a sausage. Asylum Seekers get a low level of support whilst their case is heard. If they're refused then they can leave, go 'underground' or hope that they get Exceptional Leave to Remain.
Proving which was the last country they travelled through to the point where that country will accept them is a different problem. If someone claims asylum in Luton, having entered the UK illegally, how do you prove where they arrived from? The French aren't going to accept them. What do you do then?