What is the z-domain transfer function of variable delay?
A (discrete-time) system with a variable delay must be time-varying, and for this reason it cannot be described by a transfer function of the form

$$H(z)=\sum_{n=-\infty}^{\infty}h[n]z^{-n}\tag{1}$$

where $h[n]$ is the impulse response of the system.
Only linear time-invariant (LTI) systems can be described by a transfer function as given in $(1)$.