The authors consider the quasidifferential equation \[ \delta (x(t+\Delta),\varphi (\varepsilon ,\Delta ,t,x(t)))=o(\Delta) \] in a locally compact space \(X\) with the metric \(\delta (\cdot ,\cdot)\). Here, \(\varepsilon \) is a small parameter, and for each fixed \(\varepsilon \) the mapping \(\varphi :[0,\sigma)\times [t_0, t_0+T]\times X\mapsto X\) determines a local quasimotion. The averaged equation is defined as follows: \(\delta (\xi(t+\Delta),\overline \varphi (\varepsilon ,\Delta ,t,\xi(t)))= o(\Delta)\) where \(\lim_{\varepsilon \to 0}\delta (\varphi (\varepsilon ,\Delta ,t,x),\overline \varphi (\varepsilon ,\Delta ,t,x))=0\). The authors find conditions under which for every \(\eta >0\) there exists \(\varepsilon ^0(\eta)>0\) such that for every \(\varepsilon \in [0,\varepsilon _0]\) and \(t\in[0,L]\) the following estimate holds: \(\delta (x(t,\varepsilon ,x_0),\xi (t,\varepsilon ,x_0))\leq \eta \).