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.
For example, the Christoffel symbols are given for a rotating coordinate system using polar coordinates in section 2.1.5 here:
https://arxiv.org/abs/0904.4184
They are $\Gamma^r_{tt}=-\omega^2 r$, $\Gamma^{\phi}_{tr}=\Gamma^{\phi}_{rt}=\omega/r$, $\Gamma^r_{t\phi}=\Gamma^r_{\phi t}=-\omega r$, $\Gamma^{\phi}_{r\phi}=\Gamma^{\phi}_{\phi r}=1/r$, $\Gamma^r_{\phi\phi}=-r$
So, for example, calculating the radial component of the acceleration we get:
$$m\frac{d^2 r}{d\tau^2}=f^r + m \left( \omega^2 r \frac{dt}{d\tau}^2 + 2\omega r \frac{dt}{d\tau}\frac{d\phi}{d\tau} \right)$$
Which we recognize as being the relativistic versions of the centrifugal force and the radial component of the Coriolis force