The Löwner-Kufarev equation gives a complete description of the class $S$ of all univalent holomorphic functions $f$ in the unit disk normalized by $f(0)+1=f'(0)=1$. We consider the class $S^{qc}$ of all functions from $S$ that admit quasiconformal extension to the whole Riemann sphere fixing $\infty$. There is a well known Becker's sufficient condition for the Löwner-Kufarev equation that guarantees a function from $S$ to be from $S^{qc}$. We study subordination chains of quasidisks bounded by analytic curves and corresponding motions on the modelling universal Teichmüller space. This leads to a specific form of the Löwner-Kufarev equation.