Gravity will attempt to pull the wormhole shut. But lets say you overcome that by putting some exotic matter into it so that keeps it open.

The effect of the ground level wormhole will pull at things near it on the other side. So if you say, threw a ball through it, as it leaves the one at ground level, it would start 'falling' back towards the open wormhole because the effect of gravity is affecting things on that side as well.

If it had enough momentum to break free from that side and pass downward, it still would have been slowed until the 'typical' gravity of the earth took it over.

Another, more dramatic, way of thinking about it is: If you opened one side at ground level, and another side near a black hole, the black hole would pull the earth into it.