Motivated by a longitudinal oral health study, the Signal-Tandmobiel (R) study, we propose a multivariate binary inhomogeneous Markov model in which unobserved correlated response variables are subject to an unconstrained misclassification process and have a monotone behavior. The multivariate baseline distributions and Markov transition matrices of the unobserved processes are defined as a function of covariates through the specification of compatible full conditional distributions. Distinct misclassification models are discussed. In all cases, the possibility that different examiners were involved in the scoring of the responses of a given subject across time is taken into account. A full Bayesian implementation of the model is described and its performance is evaluated using simulated data. We provide theoretical and empirical evidence that the parameters can be estimated without any external information about the misclassification parameters. Finally, the analyses of the motivating study are presented. Appendices 1-7 are available in the online supplementary materials.