Results on asymptotic normality for the maximum likelihood estimate in hidden Markov models are extended in two directions. The stationarity assumption is relaxed, which allows for a covariate process influencing the hidden Markov process. Furthermore a class of estimating equations is considered instead of the maximum likelihood estimate. The basic ingredients are mixing properties of the process and a general central limit theorem for weakly dependent variables. The results are illustrated with a cyclic model for the progesterone concentration in cowmilk.