Shouldn't a photon traversing the vacuum always be associated with a gravitational wave? In perusing the linearized Einstein equation, it appears that even a classical electromagnetic plane wave would always have to be associated with a tensor perturbation to the background spacetime.  For a wave in the "z" direction say:
$$\partial_{\alpha}\partial^{\alpha}h_{zz}=kT_{zz}
 $$
where the plane em wave has the $T_{zz}$ as the only non-zero component of the stress energy tensor and $h_{\mu\nu}$ is the perturbation to the background metric. 
From a qualitative view, they could never separate as the wave would always generate such a perturbation about it as it traverses the vacuum. To be consistent effects of an expanding universe would have to "redshift" both waves precisely the same.
Is this the case? And how come I never see it mentioned?  It seems strange or rather fundamental that electromagnetic propagation would always have to be associated with this.  
 A: The same is true for anything else moving in vacuum: it perturbs the local spacetime. Everything carries about with it such a perturbation to spacetime. And similar things arise in purely electromagnetic effects: each charged object perturbs the electromagnetic field where it is.
Contributions to the total dynamics arising from this kind of effect are generally gathered under the name "self-force". The name refers to a contribution to the net dynamics that arises from the changes created by the object under discussion. It is a second-order effect and therefore usually small. In the case of electromagnetic waves and gravity it is especially small because gravity is weak.
Now the question arises, is a beam of light really in free fall (and therefore following a null geodesic) or isn't it? The answer is that a weak beam of light is in free fall, but a sufficiently intense beam of light is not. However the gravitational effect of a beam of light on itself is such as to produce no net focusing or defocusing at first order in the metric perturbation---which is already quite an interesting observation.
Finally, the redshift associated with cosmic expansion is first and foremost a freefall effect, so it can be calculated as usual for a weak beam of light. For sufficiently intense beams some sort of correction would come in, I suspect, but I have not seen it calculated. Perhaps people working on gamma ray bursts have found that they need to carry out such a calculation.
A: I will tackle this and please, somebody with better understanding should correct me.
To start with, the title talks of "a photon traversing the vacuum" and the content is only about classical electromagnetic waves and gravitational waves. 
In the particle framework, assuming an effective quantization of gravity, one should be considering "photon graviton" interactions. In flat space the photon will be running straight without interacting . At the particle level, if there exists a curvature, the question becomes " is there a photon graviton interaction" in curved space?  From what I see  in searching there does not exist a definitive proposal even for effective quantized gravity. There are people studying this (example ), where compton like scatterings are introduced, but in no way can it be considered a standard. Certainly due to the weakness of the gravitational coupling this will be a very very  small effect except near horizons of black holes.
In the particle framework the answer is that in curved space there should be gravitons which will be exchanged with the photons , but they will be off mass shell virtual gravitons. If there is a compton type interaction with a graviton of the curvature  then one graviton will go off and the photon will lose energy.
A classical  electromagnetic wave is built up from zillions of photons, and it is supposed that also the  gravitational wave is composed by zillions of gravitons. The difference in the couplings is of order 10^-37. So for each 10^37 photons there may be a graviton produced to add up in a complicated manner to a gravitational wave derived from the  photon beam, in curved space.
The above leads me to state that  only in curved space a photon beam might generate a correlated gravitational beam  .
Let us forget photons and gravitons and take the Maxwell's equations in curved space time. Yes , there will be a stress energy tensor associated with a beam of light, but  as is true generally, both for electromagnetism and general relativity, waves are generated only through accelerations. There is a review talk here.. A plane  electromagnetic wave does not undergo acceleration in flat space just because it has a stress energy tensor.
