Social mobility and income inequality I once read an argument by the libertarian economist Thomas Sowell against the conventional wisdom that rising income is bad because most statistics which show rising income inequality do not account for individuals moving between income bands over time. 

His argument went something like this. Consider the average professional at the beginning of their career, for example a junior doctor ( the argument also seems to apply to non-professionals but in a less extreme way). They won’t be in the top income bracket when they begin working but over the course of their career they almost certainly will at some point. For this reason income inequality is not as significant an indicator of the actual economic situation for many individuals as it does not track how the incomes of individuals change, and often increase, over time. Furthermore, income inequality statistics don’t count people who don’t work, which includes a large number of wealthy pensioners. Therefore rising income inequality can even be thought of as a positive thing as it affords young people the opportunity to earn more, and thus actual leads to less actual inequality of wealth.

I couldn’t find a link to his exact words unfortunately but I suspect that he was making the point from the point of view to defend free-market capitalism.

My question is simply is is argument correct, and if not why not? 

I am a physics student, so I’m possibly just not familiar with the most technical economic theory that might explain this, but I can’t figure out why this fact (that people move between income brackets over time) is basically never addressed either in political discourse ( eg the Labour Party are always talking about income inequality but I’ve never heard the Conservatives give as eloquent a response as Sowell) or the popular literature that I’ve read ( eg newspapers on the right and left, and books by modern economists such as Ha joon Chang and David pilling).
The argument is correct

Social Mobility and Income Inequality are simply two distinct phenomenons. Mobility is arguably the more complex one, and there are many different ways to define it. As an example, absolute mobility refers to someone earning more income in absolute terms (\$ per year), while relative mobility refers to someone earning more in relative terms (going from the 60th to the 70th percentile of the income distribution). Note that the latter could occur for constant personal income, if income of all others decline. Another major distinction is made between inter- and intra- generational mobility. I would attribute part of the reason mobility is not featured often in political discource to the ambigiuous meaning of the concept. Related to this, there is no generally accepted single indicator (like the Gini coefficient for income inequality) by which mobility could be measured.

The second reason is that measuring mobility requires repeated observations of income distributions for the same individuals, ideally over a long period of time. These kind of data are still rather rare and not available for all countries in the required quality.

The scenario you have in mind ("the guy is not poor because he is likely going to earn a lot") is essentially a prospective task where one would estimate future earnings paths. While this can be done in principle, the precision of such studies are far below e.g. a detailed analysis on poverty ratios for various demographic groups.

Summing up, measuring income inequality ignores the movement of individuals between two points in time, which should be kept in mind when talking about income inequality. Povery is much more a problem if it's not a transitory, but a permanent phenomenon from an individual perspective.