We know that a small object moving fast enough can pass by a planet and escape its gravity. Would this be (theoretically) true in reverse? Meaning a planet moving fast enough past a stationary smaller object would not assert its gravitational pull on the small object? Would this mean that gravity is mitigated by velocity regardless of whether the more massive object is moving as opposed to the small object?
What you're describing is a change of co-ordinate system. For us sitting on Earth it makes sense to choose co-ordinates centred on the Earth and moving with it, because that makes the maths easy. In these co-ordinates we are stationary and the small object is moving. However it makes just as much physical sense to choose a co-ordinate system centred on the object and moving with it. In these co-ordinates the object is stationary and the Earth is moving. The key point to understand is that the physics doesn't care what co-ordinates we use. The interaction between the Earth and the object is the same in both sets of co-ordinates.
Generally in physics it's common to choose whatever set of co-ordinates makes it easier to calculate what happens. For example when the bodies are roughly equal in mass we often choose centre of mass co-ordinates i.e. we measure everything relative to the position of the centre of mass of the system. In general relativity we sometime use different co-ordinates in different parts of our system and patch them together where they join.
So, to answer your question, it makes no difference whether the object is moving and the Earth is stationary or whether the Earth is moving and the object is stationary.
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The smaller object's velocity may be high so that it can escape from falling into the planet. But, it's affected until $\infty$ because the object doesn't (can't) actually neglect the planet's gravity. So, it is deflected from its actual path. Maybe, the deflection is negligible. But, there is always a deflection based on the object's velocity and mass. In this view, the converse is also true.
Recalling SR's postulate that there's no special reference frame (i.e.) The laws of physics are the same for all inertial observers.
So, a stationary object is attracted by the planet. After the planet has passed away, you can notice the motion of the object for sure. Any body under the influence of gravitational field is affected..!