  We establish that every embedding of a Grassmann graph in a polar Grassmann
graph can be reduced to an embedding in a Grassmann graph or to an embedding in
the collinearity graph of a polar space. Also, we consider $3$-embeddings, i.e.
embeddings preserving all distances not greater than $3$, of dual polar graphs
whose diameter is not less than $3$ in polar Grassmann graphs formed by
non-maximal singular subspaces. Using the same arguments we show that every
such an embedding can be reduced to an embedding in a Grassmann graph.
