We extend the notions of finite free convolution and finite free cumulants to the setting of formal power series by introducing their natural analogues, namely $t$-deformed convolution and $t$-deformed cumulants. In this framework, we establish $t$-deformed analogues of the law of large numbers and the central limit theorem, revealing structural parallels with classical, free, and finite free probability theories. We show that the case $t=-1$ recovers classical convolution at the level of moment generating functions, thereby connecting the theory directly to classical probability. We further investigate the infinitesimal generators associated with $\boxplus^t$-continuous semigroups, deriving explicit representation formulas that clarify how these generators describe the infinitesimal evolution of the semigroup. In the case $t = d$, our results yield explicit formulas for finite free infinitesimal generators. In the case $t = -1$, we relate these generators to those of one-dimensional Lévy processes by identifying the corresponding terms in their representations. This establishes a direct connection between $\boxplus^t$-convolution semigroups and classical Lévy-Khintchine-type generators.