Zero distributions of derivatives of polynomial families centering on a set

Suppose C ⊂ℂ is compact. Let q_k be a sequence of polynomials of degree n_k →∞, such that the locus of roots of all the polynomials is bounded, and the number of roots of q_k in any closed set L not meeting C is uniformly bounded. Supposing that (q_k)_k has an asymptotic root distribution μ we provide conditions on C and μ assuring the sequence of mth derivatives (q_k^(m))_k also has asymptotic root distribution μ for any m≥ 1. This complements recent results of Totik.