  In this paper we describe braid equivalence for knots and links in a
3-manifold $M$ obtained by rational surgery along a framed link in $S^3$. We
first prove a sharpened version of the Reidemeister theorem for links in $M$.
We then give geometric formulations of the braid equivalence via mixed braids
in $S^3$ using the $L$-moves and the braid band moves.
  We finally give algebraic formulations in terms of the mixed braid groups
$B_{m,n}$ using cabling and the techniques of parting and combing for mixed
braids. We also provide concrete formuli of the braid equivalence in the case
where $M$ is a lens space, a Seifert manifold or a homology sphere obtained
from the trefoil. The algebraic classification of knots and links in a
$3$-manifold via mixed braids is a useful tool for studying skein modules of
$3$-manifolds.
