In the limit of small inertia, stratification, and advection of density, Ardekani and Stocker [Phys. Rev. Lett. 105, 084502 (2010)PRLTAO0031-900710.1103/PhysRevLett.105.084502] derived the flow due to a point-force and force-dipole placed in a linearly density-stratified fluid. In this limit, these flows also represent the far-field flow due to a towed particle and a neutrally buoyant swimming organism in a stratified fluid. Here, we derive these two far-field flows in the limit of small inertia, stratification but at large advection of density. In both these limits, the flow in a stratified fluid decays rapidly and has closed streamlines but certain symmetries present at small advection are lost at large advection. To illustrate the application of these flows, we use them to calculate the drift induced by a towed drop and a swimming organism, as a means to quantify the mixing caused by them. The drift induced in a stratified fluid is less than that in the homogeneous fluid. A towed drop induces a large drift relative to its own volume at small advection while it induces at least an order of magnitude smaller drift at large advection. On the other hand, a swimming organism induces a large partial drift as compared with its own volume irrespective of the magnitude of advection, unless the stresslet exerted by the swimmer is small. These results are useful in understanding the stratification effects on the drift-based contributions to mixing.