We propose a new class of Markov-switching (MS) models for business cycle analysis. As usually done in the literature, we assume that the MS latent factor is driving the dynamics of the business cycle but the transition probabilities can vary randomly over time. Transition probabilities are generated by random processes which may account for the stochastic duration of the regimes and for possible stochastic relations between the MS probabilities and some explanatory variables, such as autoregressive components and exogenous variables. The presence of latent factors and nonlinearities calls for the use of simulation-based inference methods. We propose a full Bayesian inference approach which can be naturally combined with Monte Carlo methods. We discuss the choice of the priors and a Markov-chain Monte Carlo (MCMC) algorithm for estimating the parameters and the latent variables. We provide an application of the model and of the MCMC procedure to data of Euro area. We also carry out a real-time comparison between different models by employing sequential Monte Carlo methods and some concordance statistics, which are widely used in business cycle analysis.