A short bibliography on Income Mobility
From an economic point of view, the topic can be introduced with two papers by Shorrocks:

A. F. Shorrocks (1976) Income Mobility and the Markov Assumption. The Economic Journal, Vol. 86, No. 343 (Sep., 1976), pp. 566-578

A. F. Shorrocks (1978) The Measurement of Mobility. Econometrica, Vol. 46, No. 5 (Sep., 1978), pp. 1013-1024

There is a third important paper by Atkinson "The Measurement of Economic Mobility (1981), but I did not manage to find it. A companion paper to it could be

Dardanoni, V. 1993. Measuring social mobility. Journal of Economic Theory 61, 372-94.

as both consider the consequences of social mobility in term of welfare (the welfarist approach). Apart from those three papers, there exist a certain number of surveys introducing the topic:

Fields, G. and Ok, E. 1999a. The measurement of income mobility: an introduction to the literature. In Handbook on Income Inequality Measurement, ed., J. Silber.
Boston: Kluwer.

Fields, G. and Ok, E. 1999b. Measuring movement of incomes. Economica 66, 455-72.

Fields, G. and Ok, E. 1996. The meaning and measurement of income mobility. Journal of Economic Theory 71, 349-77.

Fields, G., Leary, J. and Ok, E. 2002. Stochastic dominance in mobility analysis. Economics Letters 75, 333-9.

One way to formalise income dynamics is to consider a Markov process between income states with transition matrix $P$. That matrix can be constrained in various ways such as being monotone:

Valentino Dardanoni (1995) Income distribution dynamics: monotone Markov chains make light work.  Social Choice and Welfare, 12:181-192.

Monotonicity means that the expected future income is an increasing function of present income, or in other words that when starting from income state i an individual faces a better lottery than starting from state i-1. This simply means that each row of the transition matrix P is stochastically dominated by its follower.
There are complicated econometric problems when one's want to test for monotonicity (regularity). A first paper states the problem:

Valentino Dardanoni and Antonio Forcina (1998) A Unified Approach to Likelihood Inference on Stochastic Orderings in a Nonparametric Context. Journal of the American Statistical Association, Vol. 93, No. 443 (Sep., 1998), pp. 1112-1123.

together with

Valentino Dardanoni, Mario Fiorini and Antonio Forcina (2012) Stochastic monotonicity in intergenerational mobility tables. Journal of Applied Econometrics. 27: 85-107.

Income mobility is progressive if the expected income of the poor grows at a quicker pace than the expected income of the rich. This type of condition is requested in:

Benabou, R. and Ok, E. A. (2001) Social mobility and the demand for redistribution: The POUM hypothesis. The Quarterly Journal of Economics, 116(2), pp 447-487.

This second paper, however unpublished is very important to detail and compare two ways of specifying progressivity

Roland Benabou and Efe A. Ok (2001) Mobility as progressivity: Ranking income processes according to equality of opportunity, Working paper National Bureau of Economic Research.

Testing for progressivity is of course a complex task as it requires testing for a set of inequalities. The following paper is worth being read:

John P. Formby, W. James Smith, Buhong Zheng (2004) Mobility measurement, transition matrices and statistical inference. Journal of Econometrics 120, pp 181-205.