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Consider an object at rest at the Lagrange point between two large but unequal masses. Then consider that the two masses are moving towards each other due to mutual gravitational attraction. As the masses move together the central object...
... should tend towards the smaller mass as the Lagrange point moves due to the change in the intersecting fields.
It would appear to an observer at this point that the field of the smaller mass is greater than that of the larger.
Thus violating the equivalence principle.
If you are dealing with point masses, the three will collide at the same time. There is no discontinuity in the classical gravitational equations. If you are dealing with bodies of finite radius at short distances compared with the radii, the simple gravitational equations don't apply, and the Lagrange Point is not fully defined by the centers of mass. You need to know the density distribution of the bodies.
An aside: I think the question may have been using the term "Lagrangian point" rather loosely.Even though an arbitrary 3-body problem cannot be solved in closed form, Euler and Lagrange managed to find 5 solutions where a small mass would remain in a stable orbit, providing the other 2 objects were in a circular orbit.If the larger objects are in a circular orbit, then they cannot come closer together.
OK let's put it another way. Say you have a speck of dust with the same mass as 1 Plank mass and also a Planck mass black hole. take these as both a set distance above a celestial object with nil atmosphere and with the horizon of the black hole aligned with the bottom of the dust speck. As they approach the surface which one will touch it first? If the black hole beats the dust speck then an increase in density must violate the equivalence principle.