We develop a three-dimensional Brownian dynamics model to study how spherical nanoparticles (NPs) move in a flowing gel that mimics the mucus layer. The gel is represented by randomly oriented rigid fibers that translate and rotate under flow of the solvent, and interact with the NPs via steric repulsion. The NPs diffuse and also convect with the flow. We consider three external flows: a steady simple shear, and two spatiotemporally periodic cilia-driven flows representing a diseased state and a healthy state. The diseased state features a vortical flow with suppressed mucociliary clearance. The healthy state has an extra flow added onto the vortical flow that is largely parallel to the epithelium underlying the mucus layer. Under simple shear flow, the fibers align with the flow and produce anisotropic NP diffusion, with elevated streamwise diffusivity and suppressed transverse diffusivity. Both cilia-driven flow fields affect the gel structure similarly, with streamwise alignment near the epithelium, but lack of alignment near the top of the mucus layer. Their effect on NP diffusion is similar to that of the shear flow, with enhanced streamwise diffusion and suppressed transverse diffusion. The main difference between the two cilia-driven flows lies in NP clearance; the healthy flow carries the NPs away before they penetrate far into the mucus layer toward the epithelium. Thus, the model establishes a mechanistic framework for understanding NP dynamics in flowing mucus and provides quantitative guidance for the design of inhalable nanomedicine.