Summary: This paper deals with the Keller-Segel(-Navier)-Stokes system with indirect signal production \[ \begin{cases} n_t + u \cdot \nabla n = \Delta n - \nabla \cdot ( n \nabla v ) + r n - \mu n^2, \\ v_t + u \cdot \nabla v = \Delta v - v + w, \\ w_t + u \cdot \nabla w = \Delta w - w + n, \\ u_t + \kappa ( u \cdot \nabla ) u = \Delta u - \nabla P + n \nabla \Phi, \quad \nabla \cdot u = 0 \end{cases} \eqno{(\star)} \] in a bounded and smooth domain \(\Omega \subset \mathbb{R}^N\) (\(N = 2, 3\)) with no-flux boundary for \(n, v, w\) and no-slip boundary for \(u\), where \(r \in \mathbb{R}\), \(\mu \geq 0\), \(\kappa \in \{0, 1 \}\) and \(\Phi \in W^{2 , \infty}(\Omega)\). In the case without logistic source (\(r = \mu = 0\)), it is proved that for all suitably regular initial data, the associated initial-boundary value problem for the spatially two-dimensional Navier-Stokes system (\( \star\)) admits a globally bounded classical solution. This result improves and extends the previously known ones. We point out that the same result to the corresponding two-dimensional Navier-Stokes system with direct signal production holds necessarily imposing some saturated chemotactic sensitivity, logistic damping or small total initial population mass. In the case coupled with logistic source (\(r \in \mathbb{R}\), \(\mu > 0\)), it is shown that for any reasonably regular initial data, the corresponding initial-boundary value problem for the spatially three-dimensional Stokes system (\( \star\)) possesses a globally bounded classical solution, and that this solution stabilizes toward the corresponding spatially homogeneous equilibrium with the explicit convergence rates for the cases \(r < 0\), \(r = 0\) and \(r > 0\). We underline that the global boundedness of classical solution to the corresponding three-dimensional Stokes system with direct signal production was obtained only for \(\mu \geq 23\) (or sublinear signal production), and that the convergence result to the corresponding system with direct signal production was established only for \(r = 0\) and \(\mu \geq 23\). Our results rigorously confirm that the indirect signal production mechanism genuinely contributes to the global boundedness of classical solution to the Keller-Segel(-Navier)-Stokes system.