  This work identifies a class of moves on knots which translate to
$m$-equivalences of the associated $p$-fold branched cyclic covers, for a fixed
$m$ and any $p$ (with respect to the Goussarov-Habiro filtration.) These moves
are applied to give a flexible (if specialised) construction of knots for which
the Casson-Walker-Lescop invariant (for example) of their $p$-fold branched
cyclic covers may be readily calculated, for any choice of $p$. In the second
part of this paper, these operations are illustrated by some theorems
concerning the relationship of knot invariants obtained from finite type
three-manifold invariants, via the branched cyclic covering construction, with
the finite type theory of knots.
