This can be easily derived if one considers that the radial expansion takes place at the speed of light and the geometry of the figure.
For the distance between the objects to diminish, the tilt (velocity) has to be larger than the Hubble tilt (velocity) between the two objects, thus the proposed experiment doesn't make sense and that is the reason for the paradox.
For sake of simplicity, I made B to be at rest with respect to the Fabric of Space, otherwise both objects would be moving to their resting positions. Notice that this is exactly what the reader asked. This is the relative resting condition, object A has the same inclination (velocity) as it would have at position B (Hubble Velocity). This is the condition for spatial resting that is, one is at rest with respect to the other object.
Other initial conditions or kinematics can be easily inspected by realizing that any local tilt of the Fabric of Space will travel until it relaxes, which is the reason for motion or the Hypergeometrical Universe Newton's Second Law.
If the impulse is such that their paths crosses, their trajectory would be represented by the next diagram:Object A will pass by object B and will asymptotically reach the positions depicted by A' and B' when the Radius of the Universe reaches infinity or earlier if there were any dissipative process.
There both regions of the Fabric of Space will be relaxed and the objects will run away from each other at their Hubble speed. Notice that their relative speed is always their Hubble speed. The only difference is the unobservable relaxation state of the Fabric of Space.
Up to now the Q (quality factor) of the Fabric of Space has been considered to be infinite.