Effective refining of Borel coverings

Given a countable family $(\mathbf{\Gamma}_i)_{i\in I}$ of additive or multiplicative Baire classes ( $\mathbf{\Gamma}_i=\mathbf{\Sigma}^0_{\xi_i}$ or $\mathbf{\Pi}^0_{\xi_i}$) we investigate the following complexity problem: Let $(A_i)_{i\in I}$ be a Borel covering of $\omega^\omega$ and assume that there exists some covering $(B_i)_{i\in I}$ with $B_i\subset A_i$ and $B_i\in\mathbf{\Gamma}_i$ for all $i$; can one find such a family $(B_i)_{i\in I}$ in $\varDelta^1_1(\alpha)$ where $\alpha\in\omega^\omega$ is any reasonable code for the families $(A_i)_{i\in I}$ and $(\mathbf{\Gamma}_i)_{i\in I}$? The main result of the paper will give a full characterization of those families $(\mathbf{\Gamma}_i)_{i\in I}$ for which the answer is positive. For example we will show that this is the case if $I$ is finite or if all the Baire classes $\mathbf{\Gamma}_i$ are additive, but in the general case the answer depends on the distribution of the multiplicative Baire classes inside the family $(\mathbf{\Gamma}_i)_{i\in I}$.