It helps if you consider the components of the acceleration of the smaller planets due to the gravitation force of each other planet. Here is a rough diagram showing the components of acceleration for each planet, assuming the largest does not accelerate due to its large mass:
The red arrow shows the component of acceleration of a planet due to the gravity of the largest planet. The green arrow is the component due to the other planet (not the largest).
Now, consider the bodies accelerate only by the red components of acceleration (i.e. ignore the gravitation effects between the two smaller planets). As the centres of gravity for the smaller planets are the same distance away, the motion of the two smaller planets will be perfectly symmetrical, and both planets hit the largest at the same time (assume smaller planets have similar radii/size).
Now, let us add the effects of the green arrows (i.e. the gravitation effect between the smaller planets). The angle between the red arrow and the green arrow is less than 90°, so this means that the green arrow will add to the acceleration of the planet towards to largest planet. Let us assume that the green arrows are sufficiently small so that the total component of acceleration perpendicular to the line connecting each small planet to the largest planet does not cause large rotational effects about the largest planet (i.e. effects such that the small planets do not travel in approx. straight lines towards the largest planet, complicating matters, and possibly throwing the smaller planets into orbit around the larger!).
The less massive planet will have a larger green arrow, and the larger of the two green arrows has a greater contribution to the component of acceleration in the radial direction (radial, as in the line connecting the smaller planet to the biggest planet). This means that, in this instance, the less massive planet is accelerating slightly faster than the slightly more massive planet towards the biggest planet. From this, it is sufficient to assume that... drumroll the smaller planet will hit the largest planet first!
This is true for all values of d and h, assuming the values don't cause the smaller planets to collide with each other first, and that the path between each smaller planet, and the largest planet is approximately straight. (i.e. d is above a critical value)
Best Answer
In all cases, the two objects move towards one another. In fact they experience exactly the same gravitational force. However, because acceleration equals force over mass $$\mathbf{a} = \frac{\mathbf{F}}{m}$$
that equal forces causes the heavier object to accelerate much less than the lighter one. But technically, the Earth does move towards you very slightly when you jump. However, it first moves slightly away from you because in order to jump you have to push it. By the time you land it returns to its original position.
The two objects will meet at their centre of gravity. That is to say if, for example, one mass is twice as big as the other, the meeting point will be one quarter of the way from the heavy mass to the light one. In general it is the point where
$$m_1 * r_1 = m_2 * r_2$$
We can't move the Earth by jumping indefinitely because the push-away from the jump exactly cancels the pull-towards from gravity. There is no net motion. This is the same reason you can't move a boat by sitting inside and kicking the walls.