  We consider continuous-space, discrete-time Markov chains on $\mathbb{R}^d$,
that admit a finite number $N$ of metastable states. Our main motivation for
investigating these processes is to analyse random Poincar\'e maps, which
describe random perturbations of ordinary differential equations admitting
several periodic orbits. We show that under a few general assumptions, which
hold in many examples of interest, the kernels of these Markov chains admit $N$
eigenvalues exponentially close to $1$, which are separated from the remainder
of the spectrum by a spectral gap that can be quantified. Our main result
states that these Markov chains can be approximated, uniformly in time, by a
finite Markov chain with $N$ states. The transition probabilities of the finite
chain are exponentially close to first-passage probabilities at neighbourhoods
of metastable states, when starting in suitable quasistationary distributions.
