This paper studies a dynamic multinomial logit model for panel data with fixed effects. The outcome depends on lagged dependent variables, strictly exogenous regressors, and individual-specific fixed effects, with type I extreme value errors. We construct the complete set of moment conditions that are valid under a fixed number of time periods and free of the fixed effects. These moment conditions are based on a permutation-based differencing strategy that extends earlier work on binary and ordered logit models. We provide conditions under which the common parameters are identified, including regression coefficients and state dependence parameters. The results contribute to the literature on nonlinear panel models with fixed effects and short panels.Joint with Bo Honoré and Martin Weidner