The Keller–Segel–Navier–Stokes system (⋆) { n t + u ⋅ ∇ n = Δ n − χ ∇ ⋅ ( n ∇ c ) + ρ n − μ n 2 , c t + u ⋅ ∇ c = Δ c − c + n , u t + ( u ⋅ ∇ ) u = Δ u + ∇ P + n ∇ ϕ + f ( x , t ) , ∇ ⋅ u = 0 , is considered in a bounded convex domain Ω ⊂ R 3 with smooth boundary, where ϕ ∈ W 1 , ∞ ( Ω ) and f ∈ C 1 ( Ω ¯ × [ 0 , ∞ ) ) , and where χ > 0 , ρ ∈ R and μ > 0 are given parameters. It is proved that under the assumption that sup t > 0 ∫ t t + 1 ‖ f ( ⋅ , s ) ‖ L 6 5 ( Ω ) d s be finite, for any sufficiently regular initial data ( n 0 , c 0 , u 0 ) satisfying n 0 ≥ 0 and c 0 ≥ 0 , the initial-value problem for (⋆) under no-flux boundary conditions for n and c and homogeneous Dirichlet boundary conditions for u possesses at least one globally defined solution in an appropriate generalized sense, and that this solution is uniformly bounded in with respect to the norm in L 1 ( Ω ) × L 6 ( Ω ) × L 2 ( Ω ; R 3 ) . Moreover, under the explicit hypothesis that μ > χ ρ + 4 , these solutions are shown to stabilize toward a spatially homogeneous state in their first two components by satisfying ( n ( ⋅ , t ) , c ( ⋅ , t ) ) → ( ρ + μ , ρ + μ ) in L 1 ( Ω ) × L p ( Ω ) for all p ∈ [ 1 , 6 ) as t → ∞ . Finally, under an additional condition on temporal decay of f it is shown that also the third solution component equilibrates in that u ( ⋅ , t ) → 0 in L 2 ( Ω ; R 3 ) as t → ∞ .