Pushing gravity is also referred to as the Le Sage model or Majorana shielding. There are subtle differences in their models which are hard or impossible to measure.

"Pushing gravity" means that gravity is not a force that is caused by some "attractive" particle that is negotating the force between two masses, but that gravity is caused by a shower of particles or rays that hits the mass and thus moves it a little, imparting their impulse. This might explain the observations of Maurice Allais on a pendulum (like Foucault's Pendulum), which got disturbed in its normal regular motion during a solar eclipse, an effect which might have been caused by change in gravity.

Put simply, this is like a giant game of billiard:
Imagine that the entire universe outside of a body(a mass) is radiating a particle shower that pushes the body in all directions, effectively cancelling out (or if not cancelling, being unnoticeable because the entire local area is subject the same way to it). The body being hit doesn't need to absorb the impact of all waves/particles that travel through or near it, maybe it catches and shields only a portion of the particle shower. For example, the neutrino is a similar particle, most of them can just pass through Earth. Were would all those particles in the particle shower come from ? It could be a form of cosmic radiation that has been around since the Big Bang, and maybe it would be replenished by emissions, like Ettore Majorana thought, or it would decrease slowly (which would mean lower gravity in the far, far future!).

Consider that a big body like the sun is blocking out a portion of the particle shower. There is now less pressure from that direction, so Earth tries to move towards the sun, a force which we call gravity. The sun will block out a portion of the entire sky of a size that decreases with the square of the distance, so we got 1/r2; the probabilty for blocking the shower increases with the mass of the sun m1 and the probability for receiving a particle in the shower and catching its impulse is proportional to m2, the mass of Earth.

So we get gravity = (m1*m2)/r2, which is exactly The Law of Gravity normally used for two bodies since Newton.

Back to the Allais effect, so if three bodies line up, like the sun, the moon and Earth during an eclipse, the gravity of the middle body(the moon) should be smaller than expected because the other body is blocking the particle shower a bit already. So one would get the observed change in gravity.

Measuring the effect should allow calculation of the intensity/catching/shielding number of the shower, but so far(2004), this is just a theory about gravity. If you would construct an Antigravity shield, which would absorb a lot of the particles in the shower, then the shield would probably be blown away very quickly by the impact of all the particles. In effect, it would have a really high mass, so this would better be called supergravity.

In 2004, this theory was investigated and made popular by Chris Duif from Delft.

unperson says: If a mass is accelerated to a constant higher speed, it would experience a drag force in the shower of pushing gravity particles.

A reply: However, if the pushing gravity particle travels at the speed of light c, then due to Einsteins theory of relativity, the difference in velocity to any other particle is still c - you can't move faster than light. Since there is no change in relative velocity, there is no change in drag either.
However, an acceleration might cause a change in inertia caused by pushing gravity particles.
It might also turn out that this particle exists and explains the anomalies, but is not the gravity particle but causes a similar weaker force.