Summary: This paper deals with an attraction-repulsion chemotaxis-(Navier)-Stokes model with logistic source \[ \begin{cases} n_t + \mathbf{u} \cdot \nabla n = \Delta n - \chi \nabla \cdot(n \nabla c) \\ + \xi \nabla \cdot(n \nabla v) + \mu n(1 - n), \qquad &(x, t) \in \Omega \times(0, \infty), \\ c_t + \mathbf{u} \cdot \nabla c = \Delta c - c + n, &\qquad (x, t) \in \Omega \times(0, \infty), \\ v_t + \mathbf{u} \cdot \nabla v = \Delta v - v + n, \qquad & (x, t) \in \Omega \times(0, \infty), \\ \mathbf{u}_t + \kappa(\mathbf{u} \cdot \nabla) \mathbf{u} = \Delta \mathbf{u} + \nabla P + n \nabla \phi, \qquad & (x, t) \in \Omega \times(0, \infty), \\ \nabla \cdot \mathbf{u} = 0, \qquad & (x, t) \in \Omega \times(0, \infty), \end{cases}. \] under homogeneous Neumann boundary conditions in a smooth bounded domain \(\Omega \subset \mathbb{R}^N\), \(N = 2, 3\), where \(\kappa \in \{0, 1 \}\), the parameters \(\chi\), \(\xi\) and\(\mu\) are positive. This system describes the evolution of cells which react on two different chemical signals in a liquid surrounding environment. The cells and chemical substances are transported by a viscous incompressible fluid under the influence of a force due to the aggregation of cells. Firstly, when \(N = 2\) and \(\kappa = 1\), based on the standard heat-semigroup argument, it is proved that for all appropriately regular nonnegative initial data and any positive parameters, this system possesses a unique global bounded solution. Secondly, when \(N = 3\) and \(\kappa = 0\), by using the maximal Sobolev regularity and semigroup technique, it is proved that the system admits a unique globally bounded classical solution provided that there exists \(\theta_0 > 0\) such that \(\frac{\chi + \xi}{\mu} < \theta_0\). Finally, by means of energy functionals, it is shown that the global bounded solution of the above system converges to the constant steady state. Furthermore, we give the precise convergence rates.