Mappings slightly changing a fixed cross-ratio

Given a complex number $\lambda\ne0,1$, we consider local homeomorphisms of a domain $D\subset\overline{\mathbb C}$ that, in a neighborhood of every point, change slightly (with a given smallness parameter $\delta$) the cross-ratio of tetrads with fixed cross-ratio $\lambda$. We prove the quasiconformality of these mappings and obtain bounds for the coefficient of quasiconformality tending to 1 as $\delta\to0$.