In QED, can a photon travelling in free space experience change in its frequency? If we want to find the probability amplitude of a photon to travel from point A to B, do we need to consider the amplitude that its frequency may change in the path? 
Also, is it possible to detect the frequency of the photon at points A and B without disturbing the experiment, such that we only need to worry about the shift in frequency in the middle?
If so, can we still use the equation $K\propto e^{i\frac{2\pi L}{\lambda } }$ for the kernel of the motion, and integrate over all the intermediate changes in $\lambda$ to get the final amplitude?
Here, $L$ is the length of the path.
 A: No, in the path integral approach to quantum mechanics, the photon is to be assumed to have a constant frequency throughout its path. It can be assumed to take whatever path it likes, moving at the speed of light (this one appears to be only an approximation for large distances), and it can also start it's journey at any time, (which means that if a detector received a photon at a particular instance from a source some distance away from it, and if there is no way of telling when exactly the photon left from the source to complete it's journey towards the detector, then amplitude for all possibilities of the photon leaving the source at different times has to be included).
A small detail which needs to be added here is that if we are considering all the possibilities of time  for the photon to leave the source, then the initial imaginary phase of all the photons will be the same. This is necessary, and corresponds to the assumption that the source is a coherent source of monochromatic light.
This answer is based on Feynman's book QED. If there is some error, kindly point it out.
