There is no such thing as the Lorentz transform with non-constant velocity. By definition the Lorentz transform is a transform between inertial frames, so the velocity is constant. 

However, your question could be broadened to ask for arbitrary transforms, especially non inertial ones. A couple of people have mentioned the Rindler coordinates, which is the simplest such transform. Under any generic transform you can write:

$$m\frac{d^2 x^{\mu}}{d\tau^2}=f^{\mu}-m\Gamma^{\mu}_{\nu\lambda}\frac{dx^{\nu}}{d\tau}\frac{dx^{\lambda}}{d\tau}$$

This is the equivalent of your expression for any coordinate system. The $\Gamma$ terms are called the Christoffel symbols and can be calculated from the metric in any coordinates of interest