I like immigration, but it is a disaster for econometrics. People, would you please stand still while we measure you?

You are probably right that most immigrants arrive into the lower deciles. A hundred or so wealthy EB-5 newcomers won't skew this measure much.

But to be consistent, we should then assume that many of the fathers who started in the bottom 10% were themselves immigrants. I don't see any reason to think that future immigrants would be poorer than past immigrants, so if bottom-percenters of any background find themselves moving up the ladder, it is likely for the same reasons that natives advance, and not simply an artifact of poorer and poorer new arrivals.

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Hmm, it occurs to me that my argument is not so good in light of population growth. If immigrants are constantly increasing the number of people in the bottom deciles, it will tend to shift the goalposts so that most people's "place in line" improves even if their income number does not change. (On second second thought, this applies even if all the immigrants are equally poor. Also, a higher rate of reproduction among poorer deciles would contribute to this distortion.)

So you are probably wise to account for some "decile inflation" given that U.S. and Canadian policy kinda sorta welcomes poor foreigners.