How does gravity affect the wavelength of light?
A partial answer: The lower the gravitational potential the slower the clocks, see Gravitational time dilation. This means, as long as your flashlight is behind you in constant distance, it appears blue-shifted to you. With roles exchanged (light below you in constant distance) it appears red-shifted.
By the heterogeneity of the gravitational field the shift grows with the time, while falling in constant distance.

This is called the Einstein shift, which is not the same as the doppler effect. The doppler effect is caused by an objects velocity relative to the observer.