Very cool, if true

It doesn't matter if you redistribute the wealth away from the non-doms - unless you redistribute it away such that they are on the median income you are increasing inequality.

Only if you take a very crude measure (actually I have no idea which measure you have in mind that has this property) -- if you take the GINI coefficient this need not be the case, indeed it need not be nearly be the case and sometimes only a small degree of redistribution will be necessary.

Here's an examples -- pretty much first example I tried:
Initial population, income = 1.0,1.0,1.0,1.0,10.0 -- GINI = 0.514
New arrival (our hypothetical non-dom) has income 12 -- population would be
1.0,1.0,1.0,1.0,10.0,12.0 -- GINI = 0.525 (marginally worse)
Redistributive tax of only 2.0 given to poorest
Redistributed wealth
1.5,1.5,1.5,1.5,10.0,10.0 -- GINI = 0.43 (considerably improved)

Unless I made a cock up with my quick and dirty GINI calculator that is. That's not much of a redistribution really either. Of course not massively realistic but the point is you don't have to tax/redistribute much to create an improvement. Certainly not to get someone down to the median or the mean.

the immigrants do reach median (or higher) levels - but it takes a generation or so

Not in the UK where, as I mentioned earlier, social mobility is poor and getting worse. Report here is eye opening.

http://www.ucl.ac.uk/~uctpb21/Cpapers/Ethn_2gen_revision_C1.pdf