Well, we can construct artificial examples like that I guess. E.g. We kill the 100 poorest people in the country (or allow them to quietly starve) increasing the mean wealth (and decreasing inequality) or we import 100 wealthy misers who never spend anything or pay tax increasing the mean wealth.

So, "inequality is not necessarily a bad thing" in the same way that "increasing mean wealth is not necessarily a good thing".

It would be the situation if you let rich non doms into the country with no plan to try to redistribute wealth from them and you let poor immigrants into the country with no plan to redestribute wealth to them and if that decision was taken at a single moment in time. However, politics in reality does not work like that, it is an ongoing process... there's not usually a defined "start point" where "poor people enter country from date X" and even when there is, this will continue to happen so a government always has a choice to allow such policies in a way which will redistribute wealth more equitably or not.

So, while you can say "let poor people in or do not" is a valid and realistic political decision, in reality, what happens is that this is an ongoing process over many decades which can either be coupled with redistributive policies to even out the wealth imbalance or can be done in the absence of such. Unfortunately, successive governments have taken the latter route.

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