Let $X$ be a Banach space, $\mathbb{R}_+:=\left[0,\infty\right)$, and let $S:\mathbb{R}_+\times X\to X$. We assume that $S$ is continuous with respect to each variable and has the flow property, i.e. $$S(0,x)=x\;\;\mbox{and}\;\;S(t+s,x)=S(t,S(s,x)).$$
Further, let $\lambda:X\to \left[m,M\right]$ (where $0<m<M<\infty$) be a continuous function. Now consider $L:\mathbb{R}_+\times X\to \mathbb{R}_+$ given by
$$L(t,x)=\int_0^t \lambda(S(s,x))\,ds,$$
and define $H:\mathbb{R}_+\times X\to\mathbb{R}_+$ so that $t\mapsto H(t,x)$ is the inverse of $t\mapsto S(t,x)$.
I am interested in knowing if it would be possible to find some general sufficient conditions regarding $S$ and $\lambda$ under which $M:\mathbb{R}_+\times X\to X$, given by $$M(t,x)=S(H(t,x),x),$$ has the flow property.
I have observed that $H$ should satisfy $$H(t+s,x)=H(t, S(H(s,x),x))+H(s,x).$$ It holds, for instance, if $X=\mathbb{R}$, $S(t,x)=xe^{t}$ and $\lambda(x)=|x|$. In this case we have $H(t,x)=\ln(1+t|x|^{-1} )$ and $M$ is a semi-flow of the form $M(t,x)=x(1+t|x|^{-1})$.