  Strong correlations are known to severely reduce the mobility of charge
carriers near half-filling and thus have an important influence on the current
carrying properties of grain boundaries in the high-$T_c$ cuprates. In this
work we present an extension of the Gutzwiller projection approach to treat
electronic correlations below as well as above half-filling consistently. We
apply this method to investigate the critical current through grain boundaries
with a wide range of misalignment angles for electron- and hole-doped systems.
For the latter excellent agreement with experimental data is found. We further
provide a detailed comparison to an analogous weak-coupling evaluation.
