In general, one would think of the curved path of an object as accelerating and thus subject to being addressed within the context of the equivalence principle. Objects orbiting a gravitational mass are also subject to the gravitational field of that mass. This presents a special case where the gravitational "force", "pulling" the object toward the center of mass of the gravitational field, is exactly balanced by the centrifugal force, acting on the object in direct opposition to gravity. The result is that the two cancel each other out and the object experiences a free fall state.
Note, the centrifugal force is a ficticiuos force. The functional force is an object's inertial resistance to the constant and in the case of an orbit uniform acceleration toward the gravitational center of mass. This is the same as the force the object would experience were it to be under a constant and uniform acceleration along a single line of motion, and thus as per the equivalence principle, indistinguishable from the force of gravity.
So the velocity of the object in that curved path time dilates it in a manner consistent with SR, or slows it's clock accordingly.
It is important to keep in mind that while the free fall conditions resulting by a balance between the force of gravity and the centrifugal force, eliminates any time dilation due to the acceleration associated with the curved orbital path, the object is still in a location in the gravitational field where the field is weaker than it would be on the surface, of say the planet it is orbiting, so its clock will run faster than it would on the planet's surface, thus time dilated in a manner consistent with GR.
I added the philosophical disclaimer in my earlier post, because trying to explain these issues in "words" and or lay terms almost always involves analogy and some philosophical perspective or context. I was attempting to emphasize that the conceptual models described, in these posts are not exactly equivalent to the conceptual model(s) of a rigous mathematical description. An example, in lay terms I say gravity pulls objects.., while that is an accurate classical description, it is not an entirely accurate description within the context of GR. It is just the best I can do, without getting far too complex to address the basic ideas, as simply as possible.
Also keep in mind that what I am attempting to describe, winds up being a very rough sketch... An introduction that can then be improved on, by some of the more technical descriptions.
