Control Valve Position Transmitter Replacement

The position transmitter model used in some control valves (very critical) will be discounted soon. Thus, a study of the options to solve this issue by replacing it with a new sensor have started. First trials were done using a cheaper sensor to study the response of the system, but they resulted in valve hunting.
There is a list of potential root causes of this issue, but the ones I am most concerned are:

1) I presume every transmitter might have different response times. Perhaps the response of the new one is faster/slower than the speed of demand signal (tuned based in the characteristics of the old sensor), thus this mismatch is the responsible for this valve hunting. Could this be possible?

2) In such case, what would be the way to fix it? Tuning the PID again? Could it be solved adjusting the Gain of the feedback?

Some background:
-The mechanical installation of the sensor was a temporary arrangement, so maybe it was not as robust as it should be and some vibrations might have occurred.

Any advice is very welcome! :)
The device is known as a "positioner". I'd recommend not using the word 'transmitter' because that infers you're looking for an optional re-transmission signal representing the valve position back to the control system (which I don't think is the case).

"Smart" positioners have a self tuning function that tunes the response of the positioner to the dynamics of the process, which is the combination of the valve mechanics (sticktion) and the process pressures on the valve trim.

With regards to valve linkage, your test is only as good as the setup. Warped, misaligned, broken or otherwise dysfunctional linkage can have serious nagative impact on positioner/valve performance. Fix the linkage, then do your testing.

Yes, gain adjustments will make a difference in positioner performance, but that's after the linkage gets fixed. Consider leaning on your vendor to show you how well their smart positioner commissions and performs.
As per my understanding, it is a "transmitter". It only gives the position feedback to the control system, who sends the demand to the servovalves. This device does not move the valve.

Obviously, first of all, I will check the linkage to the valve. The problem is that I am pretty confident that the issue had more to be with the PID loop.
I am replacing the "feedback device" of the loop. Thus, my question is:

-Is there any sensor characteristic (such as response speed) that might be different in the new sensor and which could make the valve hunt?

-In that case, how could I tune the new device and/or the PID loop to solve the hunting?
I misread your original post thinking it was a positioner on a modulating valve. When you mentioned 'servo' I realized it is different application.

As to response rate, yes, a different PV response rate will affect the PID. The sensor's spec sheet should tell you what response the new one has, whether it is damped or not or what the output update frequency is.

Another thing to check is whether the range represented by the output of the new sensor is the same as the old sensor. Smart sensors can frequently be configured to output a fraction of the total range available.