Is there a $z$-transform like for variable sampling rate signals?
Since the powers in the $z$-transform relate to delays or lags, in the philosophy of Laurent series, I would like to mention Puiseux series, a generalization of power series with negative and fractional exponents, or the Hahn (or Hahn–Mal'cev–Neumann) series, with even more flexibility in the  exponents. However, their use in signal processing remains quite shallow to me yet. 

I suspect than rational exponents could be still tractable. Indeed, they are used on multiband ($M$-band) filterbanks, where you can spot weirdities like $X(z^{1/2})$ or $X(z^{1/M})$, but most of them are based only on rationals with $M^p$ denominators. 
More generic powers (irrational) could be way more involved. The only mentions of Puiseux series I have found on IEEE Xlore are in the domain of control and automation (definitely not my cup of tea).