In this paper, new examples of nonhomeomorphic knots and links which for certain $r$ have homeomorphic $r$-sheeted cyclic branched coverings are constructed. In particular, it is proved that the two nonhomeomorphic knots with eleven crossings and with Alexander polynomial equal to one, have homeomorphic two-sheeted branched coverings, and that knots obtained from any knot by the Zeeman construction with $p$-fold and with $q$-fold twist have homeomorphic $r$-sheeted cyclic branched coverings if $p\equiv\pm q$ $(\operatorname{mod}2r)$. The construction of examples is based on regluing a link along a submanifold of codimension 1 by means of a homeomorphism which is covered by a homeomorphism which is isotopic to the identity only through nonprojecting isotopies.