Let $A$ be an unbounded linear operator of domain $D(A)$ defined on a Banach space $X$. Suppose that $A$ generates a $C_0$-semigroup $T(t)$ which is uniformly bounded. I would like to know if there are additional assumptions under which the following conjecture is true:
For every compact subset $K$ of $X$, there exists a constant $C_K$ such that $$\mid{((T(t)-T(t'))x\mid \leq C_K\mid t-t'\mid},\quad \forall t,t'\in \mathbb{R}_+,\quad \forall x\in K.$$
If I understand your question correctly, this would mean that the function $t\mapsto T(t)x$ is Lipschitz continuous, which is equivalent for $x$ to be in the Favard space $\text{Fav}(A)$$. See for example Defintion 8.2. in these lecture notes.
If your space $X$ is reflexive, then the Favard space is exactly $D(A)$, the domain of $A$, see Corollary II.5.21 in
Engel, Klaus-Jochen; Nagel, Rainer, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics. 194. Berlin: Springer. xxi, 586 p. (2000). ZBL0952.47036.
Hence if $D(A)\neq X$ (which is the case for unbounded generators), this fails.
Unfortunately, in non-reflexive spaces, where $D(A)\subset \text{Fav}(A)\subset X$ still holds but the Favard space generally fails to be the whole space. Hence, your property cannot be fulfilled even for $K=\{x\}$ singletons if $x\notin \mathrm{Fav}(A)$.
