Transport is a central element of the PDEs found in many scientific and engineering fields, and the design and understanding of associated numerical techniques have thus drawn enormous efforts. There is now good understanding of the main distortions they produce (instabilities, smearing of discontinuities and numerical diffusion, oscillations, and loss of conservativity...) and of the main tools to control them. However, another artifact, here designated as "numerical wetting," has been somewhat neglected so far. Numerical wetting designates the low-level contamination that spreads linearly in time over all the regions of the computational domain where non-vanishing transport is present (not to be confused with usual numerical diffusion which spreads profiles sub-linearly in time). Numerical wetting may appear of marginal importance but turns out to be especially irritating in practical situations involving fluid mixing and evanescent boundary conditions, notably at "phase disappearance" episodes in multiphase flows or "wet-dry" transitions in shallow water flows. A systematic analysis of TVD limiters (Total Variation Diminishing) in the context of MUSCL transport schemes (Monotonic Upstream-centered Scheme for Conservation Laws) was recently provided [Paulin et al. 2022] and is here summarized. Main concepts here are i) the distinction between interpolant (an actual reconstructed profile y(x) within a given cell) and interpolator (a profile generating function of the field value at the cell), ii) the double monotonicity condition which should apply to interpolators, and iii) the mapping of any TVD interpolant on elliptic coordinates with respect to the two extreme TVD interpolants of first order and downwind-limited upwind [Despres and Lagoutière 2001]. Within this framework it becomes natural to continue any single-slope second-order interpolator (SS) into a slope-and-bound interpolator (SAB, instead of a slope-limited interpolator) when conflicting with mononicity and TVD conditions. As will be shown, slope-and-bound interpolators do preserve compact supports of transported functions, irrespective of their design-dependent numerical diffusion. The method is readily extended to Eulerian multi-dimensional transport with an ADI splitting (Alternate Directions). On the classical Kothe-Rider transport test (with the Super-Bee interpolator on diffuse interfaces) very substantial improvements can be observed when introducing slope-and-bound limitation.