Transform differential equation on Lie group to differential equation on representation of the Lie group Let $G$ be a connected Lie group and let $\gamma : [0,1]\to G$ be a smooth path starting at the identity $\gamma(0)=e$.
Let $R_g :G\to G$ be the right-translation by $g$, i.e., $$R_g(h)=gh.$$
Then $\gamma$ naturally satisfies a differential equation $$\gamma'(s)=[R_{\gamma(s)}]_{\ast e} X(s),\quad X(s)=[R_{\gamma(s)}]_{\ast e}^{-1}\gamma'(s)\in \mathfrak{g}\simeq T_eG.\tag{1}$$
Suppose now $U : G\to {\rm U}({\cal H})$ is a unitary representation of $G$ on a Hilbert space $\cal H$. We also know that $U$ induces a representation of the Lie algebra $\mathfrak{g}$ by means of the derived representation construction. We simply set, for $\Psi\in {\cal H}^\infty_U$ the smooth vectors,
$$D(X)\Psi=\dfrac{d}{ds}\bigg|_{s=0}U(\exp sX) \Psi.\tag{2}$$

My question is: can we translate Eq. (1) in the group $G$ into a differential equation for $U(\gamma(s))$ in the group ${\rm U}({\cal H})$?

My take is that the equation will necessarily involve the Lie algebra representation $D$ and will provide a way to "integrate $D$ to $U$ along paths in $G$".
My Attempt
My initial idea has been to use $[R_{\gamma(s)}]_{\ast e}:\mathfrak{g}\to T_{\gamma(s)}G$ to induce on $T_{\gamma(s)}G$ a Lie algebra structure and induce a representation $D_{\gamma(s)}:T_{\gamma(s)}G\to \operatorname{End}({\cal H})$ by means of $$D_{\gamma(s)}(X_{\gamma(s)})=D([R_{\gamma(s)}]_{\ast e}^{-1}X_{\gamma(s)})$$
and then apply $D_{\gamma(s)}$ to Eq. (1). The issue seems to be how to extract $$\dfrac{d}{ds} U(\gamma(s))$$
out of all of this, which we certainly need to appear if we want a differential equation for it.
 A: Since $U:G\to \operatorname{U}(\mathcal{H})$, we have $U_*:TG\to T\operatorname{U}(\mathcal{H})$, and in particular $U_{*g}:T_gG\to T_{U(g)}\operatorname{U}(\mathcal{H})$. The fact that $U$ is a representation implies that 
$$
U\circ R_g = R_{U(g)}\circ U
$$
($R$ denotes right multiplication in $G$ on the lhs, and in $\operatorname{U}(\mathcal{H})$ on the rhs). Taking the derivative at $e\in G$ on both sides, and applying to $\xi\in \mathfrak{g}\simeq T_eG$ gives
$$
U_{*g}([R_g]_{*e}(\xi)) = [R_{U(g)}]_{*I}(U_{*e}(\xi))
$$
or in more suggestive notation
$$
U_{*g}(\xi\cdot g) = U_{*e}(\xi)\cdot U(g).
$$
You can use this equation to translate the ODE $\gamma'(s) = X(s)\cdot\gamma(s)$ on $G$ into an ODE on $\operatorname{U}(\mathcal{H})$, namely
$$
\frac{d}{ds}U(\gamma(s)) = U_{*\gamma(s)}(\gamma'(s)) = U_{*\gamma(s)}(X(s)\cdot \gamma(s))= U_{*e}(X(s))\cdot U(\gamma(s)).
$$
$U_{*e}$ is what you call $D$ in your question, while $U_{*g}(\cdot)\cdot U(g)^{-1}$ is what you call $D_g$.
A: The answer by @user17945 inspired me on finding a seem-to-be rigorous approach and I've decided to outline in this answer. Comments and possible corrections are highly appreciated.
The main issue is that while $G$, being a Lie group, has a natural smooth structure, the same does not happen for ${\rm U}(\cal H)$. In fact, this is the subject of this question on which the paper "The Unitary Group In Its Strong Topology" is cited. The paper comments that endowing ${\rm U}(\cal H)$ with a suitable Lie group structure is still an open question. Something also alluded to in this Phys.SE thread.
In that case, in a rigorous scenario, we can't rely on a smooth structure on ${\rm U}(\cal H)$. Because of that we just shift a little bit the perspective and work with the (dense) subset of smooth vectors of the representation $\pi : G\to {\rm U}(\cal H)$ defined to be $${\cal H}^\infty_\pi = \{\Psi \in {\cal H} : \text{$g\mapsto \pi(g)\Psi$ is smooth}\}.$$
For that, let us fix $\Psi \in {\cal H}^\infty_\pi$ and let us define the map $\Pi_\Psi : G\to \cal{H}^\infty_\pi$ to be $$\Pi_\Psi(g)=\pi(g)\Psi.$$
Take now $\gamma : [0,1]\to G$ and consider $\Pi_\Psi(\gamma(s))$. We differentiate it to obtain $$(\Pi_\Psi\circ \gamma)'(s)=[\Pi_\Psi]_{\ast\gamma(s)}\gamma'(s).$$
Next we use $\gamma'(s) = [R_{\gamma(s)}]_{\ast e}X(s)$ to obtain $$(\Pi_\Psi\circ\gamma)'(s)=[\Pi_\Psi]_{\ast \gamma(s)}[R_{\gamma(s)}]_{\ast e}X(s) = [\Pi_\Psi\circ R_{\gamma(s)}]_{\ast e}X(s).$$
Now we should be worried with the understanding of the pushforward $[\Pi_\Psi\circ R_{\gamma(s)}]_{\ast e}:\mathfrak{g}\to \cal{H}^\infty_\pi$. We understand it from the kinematical viewpoint. This means that to make it act on $Z\in \frak g$ we take some differentiable curve $\sigma : (-\epsilon,\epsilon)\to G$ with $\sigma(0)=e$ and $\sigma'(0)=Z$ and define $$[\Pi_{\Psi}\circ R_{\gamma(s)}]_{\ast e}Z = \dfrac{d}{d\lambda}\bigg|_{\lambda =0}[\Pi_{\Psi}\circ R_{\gamma(s)}](\sigma(\lambda)).$$
We can take such $\sigma(\lambda) = \exp \lambda Z$. Then working through the definitions we find $$\Pi_\Psi\circ R_{\gamma(s)}(\exp \lambda Z) = \Pi_\Psi((\exp \lambda Z)\gamma(s)) =\pi(\exp \lambda Z)\pi (\gamma(s)) \Psi.$$
Now since $\Psi\in {\cal H}^\infty_\pi$ so is $\pi(\gamma(s))\Psi$ for any fixed $s\in [0,1]$. Differentiating at $\lambda =0$ we have just the definition of the derived representation and find $$[\Pi_\Psi\circ R_{\gamma(s)}]_{\ast e}Z = d\pi(Z)[\pi(\gamma(s))\Psi].$$
Finally we do this for $Z = X(s)$. Since we do this for each $s\in [0,1]$ fixed everything is fine and we obtain $$(\Pi_\Psi\circ \gamma)'(s)=d\pi(X(s))[\pi(\gamma(s))\Psi].$$
Reorganizing and using the definition of $\Pi_\Psi$ we find $$\dfrac{d}{ds} \pi(\gamma(s))\Psi = d\pi(X(s))\pi(\gamma(s)) \Psi.$$
Since this differential equation is valid in all of $\cal{H}^\infty_\pi$ and this is a dense subspace we may simply write, assuming extension by continuity, $$\dfrac{d}{ds} \pi(\gamma(s))=d\pi(X(s))\pi(\gamma(s))$$
