We use results by Chenevier and Hansen to interpolate the classical Jacquet-Langlands correspondence for Hilbert modular forms, which gives an extension of Chenevier’s results to totally real fields. From this, in the case of totally real fields of even degree, we obtain isomorphisms between eigenvarieties attached to Hilbert modular forms and those attached to modular forms on a totally definite quaternion algebra. More generally, for any field and quaternion algebra, we get closed immersions between certain eigenvarieties associated to overconvergent cohomology groups. Using this, we compute slopes of Hilbert modular forms near the centre and near the boundary of the weight space and prove a lower bound on the Newton polygon associated to the Up operator. Near the boundary of the weight space we give evidence that the slopes are given by unions of arithmetic progressions and we give a conjectural recipe for generating slopes.