@PeterKampf's answer is (as usual) perfectly fine in and of itself.
I would add another way to look at it, more focused on the vortices you were interested in. One way of looking at the airflow is to imagine it is simply made of two opposite vortices.
Here is how it starts: the downash created by the wing provokes the creation of two wingtip vortices, easy to see in this famous image:
Now you can clearly see the vortices are:
- Opposite (left is clockwise, right is anticlockwise)
- Of the same strength (they hold the same quantity of rotational energy from the moving air) by symmetry
- And you may imagine they are have infinite reach (though they are stronger at their respective center, and weaker far away until their effect vanishes).
Now the key is that they both interact with each other (vertices are mathematically additive):
- The left one imposes an (almost) uniform downward movement of all the air far to its right (including the right wortex)
- And vice-versa, the right vortex imposes an (almost) uniform downward movement of all the air far to its left (including the left wortex).
Of course both impose a downward movement to the air in between (that's the downwash itself).
The combined effect is at T = T+1, each vortex will have translated the other some distance down. They will then be in the same configuration again (two opposite vortices, side by side and of the same strength), and so this keeps on going until they interact with the ground or dissipate into friction (viscosity).
This is just an other way of looking at the same thing as a vortex persistence is still the consequence of inertia. Also this schematic may help visualize flow speeds induced by each vortex (remember both speed flows are additive and must be superimposed:
Edit: It is all based on the lifting line theory, particularly the horseshoe vortex. I've found few good videos of the swirl interactions. Here is one.
Since the two vortices are a full wingspan away, you can consider flow induced by the left vortex in the vincinity of the right vortex to be uniformly down. Speed is orthogonal to the offset to the center, so speed is straight down. That's not the case everywhere: above the vortex, speed is inwards. Below, its outwards. Speed beyond the wingspan is upwards. But those speeds in those regions are of no interest to the opposite vortex.
By the way, smokers can generate vortex rings. Those are the same phenomenon, just with cylindrical symmetry instead of planar symmetry. Or if you don't smoke, you can do those in a swimming pool (Yes, it's violin time!)