A function $f : S \times \mathbb{R} \rightarrow S$ could have the property
$$ f(x, a + b) = f(f(x, a), b) .$$
For example, with $S = \mathbb{R}$ it is true of $f(x, a) = x + ka$ and $f(x, a) = xk^a$.
What can this property be called? It seems like it should be something “… linear …” but I don't know exactly what.
The particular case I am immediately interested in describing using this term is that the property almost holds for the function $$f(x, a) = (1 + ka)x$$ in which case we find that $$ \begin{align} f(f(x, a), b) &= (1 + kb)(1 + ka)x \\ &= (ka + kb + k^2ab + 1)x \\ &= (1 + k(a + b + kab))x \\ &= f(x, a + b + kab) \end{align} $$ and we can then consider whether the difference $kab$ is small enough that $f(f(x, a), b)$ is a suitable approximation for $f(x, a + b)$ in the particular application.
For each $t\in \mathbb R$, you obtain a map $f_t= f(-,t): S\longrightarrow S$. Your condition means that
$$f_{s+t}(p) = f(p,s+t) = f(f(p,t),s) = f_t(f_s(p))$$
that is $f_{t+s}=f_t\circ f_s$. Observe, moreover, that in your examples we also have that $f_0$ is the identity, so by the above $f_t$ is bijective with inverse $f_{-t}$.
This means there is a map $\mathbb R\longrightarrow \operatorname{Homeo}(S)$ (where $\operatorname{Homeo}$ stands for the group of homeomorphisms of the space $S$) that is a morphism of groups. Such maps are termed one parameter groups, and arise naturally when considering flows of vector fields on manifolds.