We know there is no aether, so what is being dragged in frame dragging? I have read this question:
In stellar frame dragging what is the 'frame'?
There are several questions on this site about frame dragging, all of them take the frame that is being dragged as an axiom, they do not explain what is exactly being dragged.
None of these answer my question.
Today, we do know that there is no aether (based on the Michelson-Morley experiment).

Frame-dragging is an effect on spacetime, predicted by Einstein's general theory of relativity, that is due to non-static stationary distributions of mass–energy. A stationary field is one that is in a steady state, but the masses causing that field may be non-static, rotating, for instance. ...
They predicted that the rotation of a massive object would distort the spacetime metric, making the orbit of a nearby test particle precess.

https://en.wikipedia.org/wiki/Frame-dragging
Based on these, it is the frame/metric that is being dragged, but none of these elaborate on what the frame itself is. This could be the gravitational field that is being dragged, or the fabric of spacetime itself.
Question:

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*In frame dragging, what is exactly being dragged, the gravitational field or the fabric of spacetime itself?

 A: 
In frame dragging, what is exactly being dragged, the gravitational field or the fabric of spacetime itself?

Suppose that we have a set of three orthogonal gyroscopes and three orthogonal linear accelerometers, also called an inertial measuring unit (IMU), and that they are all exquisitely precise. Suppose further that we mount a telescope to this IMU and ensure that the telescope neither rotates nor accelerates according to the local IMU measurements. 
Now, let's place one of these telescopes in orbit around a non-spinning mass and another in orbit around an identical spinning mass. If we look carefully through the telescopes we will see that both telescopes exhibit parallax (nearby stars will shift relative to the distant stars). However, in addition to the parallax, the telescope around the spinning mass will also precess slightly (the whole star field will move in a circle). This does not occur with the telescope around the non-spinning mass.
This effect is called frame dragging. It is not predicted by Newtonian gravity but it is predicted by General Relativity, and has been measured (though not to great precision) with Gravity Probe B.
As far as "what is being dragged", the IMU experimentally defines a local inertial reference frame. That is what is being dragged. Local inertial reference frames are dragged so as to change their orientations with respect to distant objects. 
I dislike the whole "fabric of spacetime" phrase, so I would never describe something that way and “dragging the gravitational field” is ambiguous since there are multiple things that could qualify as the gravitational field in GR (the metric, the Riemann curvature tensor, the Christoffel symbols). So I would simply describe frame dragging as the dragging of local inertial frames and, if asked what that means, would describe it experimentally as I did above. 
A: I divide this answer in three section, I, II, and III, but I give them out of order: I, III, II. The content of section II precedes the content of section III logically, but you may already have the background knowledge that is discussed in section II
Section I
In order to address that question I must first get something out of the way. In terms of GR the gravitational field and spacetime are one and the same thing, as a matter of principle.
Elaborating on that 'are one and the same thing':
The spacetime of GR is a participant in the physics taking place, as expressed by John Archibald Wheeler: "Spacetime tells matter how to move; matter tells spacetime how to curve." The purpose of that phrase is to express that in terms of GR spacetime is thought of as an entity with physical properties. It's about recognizing a reciprocity: spacetime acts upon, and is being acted upon.
Before the introduction of GR the unquestioned assumption was that spacetime and gravitational field are distinct, just as spacetime and electromagnetism are distinct. In terms of GR the concepts 'spacetime' and 'gravitational field' are unified.
Now, we don't have a word for this unified concept, which is awkward. Discussions of this topic tend to switch between 'spacetime' and 'gravitational field', suggesting a distinction that isn't there.
For the remainder of this answer I will use the expression 'GR spacetime'. GR spacetime then stands for 'the gravitational field as described by the GR equations'.
To emphasize what spacetime isn't: spacetime doesn't have parts that can be tracked through time. There is no such thing as assigning a position vector to any part of GR spacetime; there is no such thing as assigning a velocity vector to any part of GR spacetime.
That is a fundamental difference with the concept of an Aether. The whole point of an Aether theory is that such an Aether can be tracked through time.
This already addresses part of your question: I surmise you are in part asking: "Hey, there's no Aether, surely that means that there isn't anything that can be dragged at all." 
(Very much an aside: it is possible that some future theory, a successor to GR, will reinstate distinction between 'spacetime' and 'gravitational field', but at present GR is what we have.)

Section III
About the expression 'frame dragging'.
Instead of 'frame dragging' this phenomenon could also have been called 'frame reorientation'.
The expression "frame dragging" is awkward to the extent that in normal use the word 'dragging' comes with a notion of dragging an object or a fluid from one position to another. Obviously in terms of GR spacetime that doesn't apply. There is no such thing as assigning a position vector to any part of GR spacetime; there is no such thing as assigning a velocity vector to any part of GR spacetime. The point is, while such vectors can't be assigned, it is still possible to make specific statements about the geometry of spacetime. See for example the two answers to the stackexchange question. How much does the curvature of space change the volume of Earth by?  That is an answerable question! That demonstrates it is possible to make very specific statements.
So my recommendation would be to translate the expression 'frame dragging' to 'frame reorientation'. The word 'dragging' comes with bagage you don't need. The thing that is indispensible is that you do think of GR spacetime as having physical properties: in the vicinity of a spinning celestial body the spacetime has the property (due to the vicinity of a spinning celestial body) that the orientation of the spin axis of a spinning gyroscope undergoes change.

Section II
The gravity Probe B mission did find corroboration of frame dragging, but around the Earth the effect is so small that even for the most sensitive equipment it is still barely detectable.
In the following section I give background information on measuring change of orientation.
When a gyroscope is spinning its spin axis will remain in the same orientation with respect to the local inertial coordinate system. That is, a force is required to make the spin axis of a spinning gyroscope change orientation. In the absence of any such force the spin axis will remain in the same orientation. 
(Keeping track of the orientation of the spin axis of a spinning gyroscope is only one way of tracking orientation. There are other ways of keeping track of orientation. It can also be accomplished with light. There is a widely used type of change-of-orientation sensor that is called fiber optic gyroscope.
The general name for a sensor array that keeps track of any form of acceleration (thus including change of orientation) is IMU, Inertial Measurement Unit. The frame dragging effect is fundamental and any IMU device is subject to it, not just gyroscopes. I use gyroscopes in this answer because a gyroscope is something we can see with our very eyes.)
According to GR, when a spinning gyroscope is in the vicinity of a spinning celestial body there will be a measurable change of the orientation of the spin axis, changing at a constant rate. That change is with respect to a more global coordinate system.
