Let \(\mathbb{S}_1\) and \(\mathbb{S}_2\) be metric spaces and let \(\Psi(\cdot|\cdot)\) be a stochastic kernel (also known as a transition probability); \(\psi\) is said to be weakly continuous if \(\psi(\cdot| s^{(n)})\) converges weakly to \(\psi(\cdot| s)\). Let \(\mathbb{S}_1\), \(\mathbb{S}_2\) and \(\mathbb{S}_3\) be Borel subsets of Polish spaces; if \(\psi(A,B| s_3):=\psi(A\times B|s_3)\), the stochastic kernel \(\psi\) is said to be semi-uniform Feller if, for every sequence \(s_3^{(n)}\) that converges to \(s\) in \(\mathbb{S}_3\) one has, for every bounded continuous function \(f\) on \(\mathbb{S}_1\) \[ \lim_{n\to\infty}\,\sup_{B\in\mathcal{B}(\mathbb{S}_2)}\,\left|\int_{\mathbb{S}_1}f(s_1)\psi(ds_1,B| s_3^{(n)})- \int_{\mathbb{S}_1}f(s_1)\psi(ds_1,B| s_3)\right|=0\,. \] The authors give different equivalent definitions of this property and show its preservation under integration; they also give a motivation from the theory of Markov decision processes with incomplete information.