An interesting insertion result about the existence of "intermediate" set-valued mappings between pairs of given set-valued mappings was obtained by Nepomnyashchii. His construction was for a paracompact domain, and he remarked that his result is similar to Dowker's insertion theorem and may represent a generalisation of this theorem. In the present paper, we characterise the $\tau$-paracompact normal spaces by this insertion property and in the special case of $\tau=\omega$, i.e. for countably paracompact normal spaces, we show that it is indeed equivalent to Dowker's insertion theorem. Moreover, we obtain a similar result for $\tau$-collectionwise normal spaces and show that for normal spaces, i.e. for $\omega$-collectionwise normal spaces, our result is equivalent to the Katětov-Tong insertion theorem. Several related results are obtained as well.