The purpose in this paper is to design finite-volumes schemes on structured grids for the transport of piecewise-constant functions (typically, indicator functions) with as low diffusion as possible. We first propose an extension of the so-called Lagrange-projection algorithm, or downwind scheme with an Ultrabee limiter, for the transport equation in one space dimension with a non-constant velocity; as its constant velocity counterpart, this scheme is designed to capture the discontinuities separating two plateaus in only one cell, and is referred to as "anti-diffusive". It is shown to preserve the bounds of the solution. Then, for two and three dimensional problems, we introduce a conservative alternate-directions algorithm, an show that this latter enjoys a discrete maximum principle, provided that the underlying one-dimensional schemes satisfy a property which may be seen as a flux limitation, possibly incorporated a posteriori in any explicit scheme. Numerical tests of this alternate-directions algorithm are performed, with a variety of one-dimensional embedded schemes including the anti-diffusive scheme developed here and the so called THINC scheme. The observed numerical diffusion is indeed very low. With the anti-diffusive scheme, the above-mentionned a posteriori limitation is necessary to preserve the solution bounds, but, in the performed tests, does not introduce any visible additional diffusion.