In this paper, we discuss the convergence of solutions to a class of parabolic chemotaxis model combined with the Navier-Stokes equations { ∂ t n ε + u ε · ∇ n ε = ∆ n ε − ∇ · ( n ε S ( x , n ε , c ε ) · ∇ c ε ) − b n ε l ε ∂ t c ε + u ε · ∇ c ε = ∆ c ε − c ε + n ε , ∂ t u ε + κ ( u ε · ∇ ) u ε = ∆ u ε + ∇ P ε + n ε ∇ ϕ , ∇ · u ε = 0 to solutions of the parabolic-elliptic counterpart formally derived in the limit as ε↘0. Let N≥2, Ω ⊂ R N is a smoothly region in physical space. For l ∈ ( 0 , N N − 1 ) , under suitable assumptions, we establish a general result that guarantees certain strong and pointwise convergence properties, provided that the assumed bounds on ∇ c ε and u ε hold in L λ ( ( 0 , T ) ; L q ( Ω ) ) and L ∞ ( ( 0 , T ) ; L r ( Ω ) ) , separately, for some λ∈(2 ,∞], q>N and r>N satisfying 1 λ + N 2 q < 1 2 .