Multistable Lévy motions are extensions of Lévy motions where the stability index is allowed to vary in time. Several constructions of these processes have been introduced recently, based on Poisson and Ferguson-Klass-LePage series representations and on multistable measures. In this work, we prove a functional central limit theorem for the independent-increments multistable Lévy motion, as well as of integrals with respect to these processes, using weighted sums of independent random variables. This allows us to construct continuous approximations of multistable Lévy motions. In particular, we prove that multistable Lévy motions are stochastic Hölder continuous and strongly localisable.