We study the proof theory of Sacchetti's modal logics, a family of logics generalizing Gödel--Löb provability logic by replacing transitivity with n-transitivity. We make three main contributions. First, we solve an open problem of Iwata by providing an effective cut elimination procedure for Sacchetti's logics. Second, building on this result, we introduce a new non-wellfounded sequent calculus for this family of logics with an improved subformula property. Third, using this calculus together with interpolation templates, we prove that Sacchetti's logics have the uniform Lyndon interpolation property, substantially strengthening previous interpolation results for these logics.