An X-ray tube creates a broad spectrum of X-rays due to Bremsstrahlung radiation and specific frequencies due to X-ray fluorescence. There's an example spectrum on the Wikipedia page about X-ray tubes. I assume you're talking about the fluorescence peaks since you refer to characteristic X-rays.
The fluorescence peaks are generated because the incident electrons collide with and eject electrons from the inner orbitals of the atoms in the target leaving the atoms ionised. When the ion recombines with an electron the energy of recombination is emitted as a photon. The energy of the emitted photon depends only on the energy of the atomic orbital that the electron is relaxing into, so it's a property of the atom. As long as the incident electrons have enough energy to ionise the atom the emitted photon will have the same energy.
So changing the energy of the incident photons may well change the intensity of the fluorescence peak, but it will not change the wavelength. Typically a commercial X-ray tube, e.g. for use in a diffractometer, will have the electron energy tuned to maximise the intensity of the required fluorescence peak (usually the $K^\alpha$.