$R_i$-congruence is defined for partial $n$-ary groupoids as a generalization of right congruence of a full binary groupoid. It is proved that for any $i$ the $R_i$-congruences of a partial $n$-ary groupoid $G$ form a lattice, where the congruence lattice of $G$ is not necessary a sublattice. An example is given, demonstrating that the congruence lattice of a partial $n$-ary groupoid is not always a sublattice of the equivalence relations lattice of $G$. The partial $n$-ary groupoids $G$ are characterized such that for some $i$, all the equivalence relations on $G$ are its $R_i$-congruences.