Let the analogous prediction for the global frame be this: any particle below 1 km above sea level must fall inexorably toward r = 0. Let your first probe be launched just above the 1 km mark. It doesn't need to be escaping, just always moving away from the Earth during our experiment. Let the second probe be launched just below the 1 km mark. As measured in any LIF containing both particles, they'll recede from each other.

On thinking this over, I realized that even in this "skydiver" LIF, you can, in fact, set up initial conditions so that the two probes are converging, even though one probe's r coordinate is increasing and the other's is decreasing. Here's how:

At time t = minus epsilon in the LIF, the "skydiver", who is at rest in the LIF, launches the first probe. At that instant, his downward velocity, relative to an observer who is "hovering" at constant global radial coordinate r, is v1. That means that, in the skydiver LIF, an object with constant r that passes through coordinates x = 0, t = minus epsilon has velocity v1 in the positive x direction at that instant. So the skydiver launches the first probe in the positive x direction with velocity v1 + a, where a is some small constant; that means the first probe's r coordinate is increasing.

At time t = 0 in the LIF, the skydiver passes the 1 km mark.

At time t = plus epsilon in the LIF, the skydiver launches the second probe. At that instant, his downward velocity, relative to an observer who is "hovering" at constant global radial coordinate r, is v2, and v2 > v1 (because, relative to observers who are "hovering" at constant r, the skydiver is accelerating downward). So, relative to the LIF, an object with constant r that passes through coordinates x = 0, t = plus epsilon has velocity v2 in the positive x direction at that instant. So the skydiver launches the second probe in the positive x direction with velocity v2 - b, where b is some small constant; that means the second probe's r coordinate is decreasing.

Now all we have to do is choose a and b so that v2 - b > v1 + a; i.e., the velocity of the second probe, relative to the skydiver (and therefore relative to the LIF) is larger than that of the first probe. (This is always possible because v2 > v1, as above.) That means the two probes will be converging, not diverging; and yet the first probe's r coordinate is increasing while the second probe's r coordinate is decreasing.

Notice the key facts that make the above possible:

(1) The skydiver is falling downwards, with respect to the Earth, with nonzero velocity. That means an object can be moving in the positive x direction in the LIF but still be falling downwards, as long as it's falling slower, with respect to the Earth, than the skydiver.

(2) The skydiver's downward velocity, relative to observers "hovering" at constant r, increases as he falls. That is what makes v2 > v1, and thus "makes room" for the second probe's velocity, relative to the LIF, to be larger than the first probe's, so that the two converge.

You may object: but that's tidal gravity, isn't it? No, it isn't; it's just downward acceleration. The above argument holds even if the skydiver's downward acceleration, relative to observers "hovering" at constant r, does not change (which of course it can't within the LIF, since tidal gravity is by definition negligible within the LIF).

Note also that none of the above changes what I've said before; all the things I said about how the relationship between global and local is very different for the astronaut LIF as compared to the skydiver LIF are still true. But perhaps the above will help show how, even in a highly non-relativistic case (all the velocities in the above example are very small compared to the speed of light), the relationship between the global r coordinate and the local coordinates in a free-falling LIF is not quite what you might think it is.