The taxis system(⋆){ut=Δu−χ∇⋅(u∇v)+uv−ρu,vt=DΔv−ξuv+μv(1−αv), is considered in bounded convex domains Ω⊂Rn, n≥1, where D,χ and ξ are positive but ρ,μ and α are merely assumed to be nonnegative. This particularly includes the special case ρ=μ=0 of a nutrient taxis system with food-dependent cell proliferation, but beyond that the system (⋆) for general parameter choices is also used as a model for Lotka–Volterra-type interaction involving prey taxis.It is firstly shown that if n≤5, then for all suitably regular initial data an associated no-flux initial-boundary value problem admits a global weak solution. To the best of our knowledge, this inter alia provides the first result on global existence in a system of the form (⋆) in a spatially three-dimensional setting when arbitrarily large initial data and parameters are involved.Secondly, under the additional hypotheses that n≤3, ρ=0 and μ<16Dαχ2 it is seen that each of these solutions becomes eventually smooth and stabilizes toward a spatially homogeneous equilibrium in the sense that(0.1)u(⋅,t)→u∞in L∞(Ω)andv(⋅,t)→0in L∞(Ω) as t→∞, where u∞ is a constant satisfying u∞≥1|Ω|∫Ωu0.This on the one hand shows that in comparison to the well-understood simple nutrient taxis system obtained on letting ρ=μ=0 and removing the proliferation term +uv in (⋆), the introduction of the latter does not substantially affect the tendency of the system to support relaxation into spatial homogeneity. Apart from that, (0.1) complements previous findings on convergence to equilibria in prey-taxis systems of type (⋆), which seem to exclusively concentrate on cases in which unlike in (0.1) the constant limit functions are a priori known, and in which thereby the corresponding asymptotic analysis is apparently simplified to a considerable extent.