Mapping transient characteristics from S domain to Z domain

Question

The $s$ and $z$ domains are linked by the expression $z = e^{sT}$, where $s$ is the Laplace variable and $T$ is the sampling period. However, I have found no rigorous method that attempts to derive all the well-developed characteristics for the LTI continuous systems such as absolute stability, relative stability, damping, overshoot, etc for the discrete systems in the $z$ too. In other words, while complex variables in the $s$ plane are mathematically mapped into the $z$ plane by the mentioned expression, how can we prove that the transient system characteristics are mapped in a similar manner; particularly knowing that there is no bijective mapping of the system poles and zeros from $s$ domain to $z$ domain. Can anyone give me some insight please?