Local constancy of dimensions of Hecke eigenspaces of automorphic forms

We use a method of Buzzard to study p-adic families of Hilbert modular forms and modular forms over imaginary quadratic fields. In the case of Hilbert modular forms, we get local constancy of dimensions of spaces of fixed slope and varying weight. For imaginary quadratic fields we obtain bounds independent of the weight on the dimensions of such spaces.