  We present the results of muon-spin relaxation ($\mu^{+}$SR) measurements on
the kagome staircase compounds Ni$_{3}$V$_{2}$O$_{8}$ and
Co$_{3}$V$_{2}$O$_{8}$. The magnetic behavior of these materials may be
described in terms of two inequivalent magnetic ion sites, known as spine sites
and cross-tie sites. Our $\mu^{+}$SR results allow us to probe each of these
sites individually, revealing the distribution of the local magnetic fields
near these positions. We are able not only to confirm the magnetic structures
of the various phases proposed on the basis of bulk measurements but also to
give an insight into the temperature evolution of the local field distribution
in each phase.
