What actually happens? It mostly depends on the pump rate, and on the size of the container. Are we discussing 6ft glove box? A 10mL sample tube?
But also, what's your goal here? A slow mixing? Or fast replacement of air? If you want to speed up the replacement of O2 by pure N2, ask that question instead.
Are you injecting nitrogen at mL/hour rates, where the system is completely dominated by diffusion? No? (That's what other answers here assume.)
If instead you're filling the volume quickly, and the gas velocity across your container is cm/sec range (so flow >> diffusion,) then the answer depends on nitrogen and air densities (temperatures,) and upon vertical position of the leak and the feed, and whether your N2 outlet is a small open tube versus a huge pipe stuffed with rags. If the N2 is injected at high velocity from a small hose, then it will form a very NON-laminar jet which entrains and mixes the existing contents of the container. In that case the O2 fraction will decrease slowly, exponentially, except for O2 trapped near corners and crevices, and near the surfaces in boundary layer.
Instead of a turbulent jet, perhaps you'd rather produce a radial laminar flow, where the wave of pure N2 pushes the air ahead of it? If so, then wrap the end of your N2 hose with several layers of cloth, or, form a bag from paper toweling. The turbulent jet is gone, and the N2 now exits uniformly from the surface of this "laminar-izer."
Are you injecting icy-cold nitrogen from a regulator, injected into warm environment? Or, warm nitrogen from a zeolite generator? N2 is a bit less dense than N2O2, but not if its temperature is low. Depending on the relative density, the N2 will either pool at the bottom of your container, or rise to the top. If you're trying to rapidly sweep out the air, then cold dense N2 should be injected from the bottom, with any exit leak-hole at the top. For warm N2, swap the direction.
Finally, if you're trying to rapidly sweep the air out of a large container, then a box or cylinder shape will temporarily retain air as vortices in the corners. To improve things your container should be a sphere, or a cylinder with hemispherical ends. Inject laminar N2 flow through the wall at the center of one hemisphere, with a leak-hole punched through the center of the second hemisphere. As long as the forwards N2 flow rate is large enough that the backwards air diffusion is insignificant, then except for some boundary-layer effects, the growing "pool" of pure N2 will push the air ahead of it.
PS, when in doubt, why not just measure? Find someone with an oxygen detector meter to borrow. Stick the probe tube in your container, measure the N2/O2 percentage versus time.