In this paper we study asymptotic properties of different data-augmentation-type Markov chain Monte Carlo algorithms sampling from mixture models comprising discrete as well as continuous random variables. Of particular interest to us is the situation where sampling from the conditional distribution of the continuous component given the discrete component is infeasible. In this context, we advance Carlin & Chib's pseudo-prior method as an alternative way of infering mixture models and discuss and compare different algorithms based on this scheme. We propose a novel algorithm, the Frozen Carlin & Chib sampler, which is computationally less demanding than any Metropolised Carlin & Chib-type algorithm. The significant gain of computational efficiency is however obtained at the cost of some asymptotic variance. The performance of the algorithm vis-à-vis alternative schemes is, using some recent results obtained in Maire et al. (Ann Stat 42: 1483-1510, 2014) for inhomogeneous Markov chains evolving alternatingly according to two different π∗-reversible Markov transition kernels, investigated theoretically as well as numerically