If an electron is moving towards the oncoming wavefronts of a light source, does it scatter/emit light at a higher frequency?

I was hoping to see if my understanding/interpretation of scattering is correct or not.

If you have an electron at rest, and it experiences a disturbance in the electric field caused by an X-ray, it will oscillate up and down. My understanding is that an accelerated charge will then emit its own virtual X-rays with the same frequency (lets assume elastic scattering) as the incident x-ray.

But in the same way we can tell whether a star is moving away from us or toward us by its doppler shift, I was wondering if that same electron was not at rest, but rather was moving TOWARDS the light, if it would be accelerated up and down at a much higher frequency (because it hits a new wavefront faster than it would if it was just sitting still).

So, a microwave will elastically scatter to a microwave on a still object. But could a microwave scatter to a gamma ray on an object that was moving fast enough towards the wavefronts?

Thanks!