The coherent structure of the state is not preserved by a general quantum evolution.
It is preserved only in very special cases, i.e. when the evolution is generated by the second quantization of a one-particle operator. For quantum mechanical systems, this amounts only to the evolution generated by the harmonic oscillator.
Let us recall that coherent states are most conveniently studied in second quantization formalism. Let $\mathscr{H}$ be the one-particle Hilbert space. Then the (symmetric) Fock space over $\mathscr{H}$ is defined as $$\Gamma_\mathrm{s}(\mathscr{H})=\bigoplus_{n=0}^\infty \underbrace{\mathscr{H}\otimes_{\mathrm{s}}\mathscr{H}\dotsm \otimes_{\mathrm{s}} \mathscr{H}}_{n}=:\bigoplus_{n=0}^\infty \mathscr{H}_n\;,$$ where $\otimes_\mathrm{s}$ stands for the symmetric tensor product, and for $n=0$ one takes the complex numbers in the direct sum.
In quantum mechanics, one takes $\mathscr{H}=\mathbb{C}$, and thus $$\Gamma_\mathrm{s}(\mathbb{C})=\bigoplus_{n=0}^\infty \mathbb{C}\cong L^2(\mathbb{R})$$ (identifying each complex space in the direct sum as the span of a Hermite function).
In the second quantization formalism, the creation and annihilation operators are well-known, and the coherent state is usually written as $$\lvert\alpha\rangle = e^{a^*(\alpha)-a(\alpha)}\Omega\; ,$$ where $\alpha\in \mathscr{H}$, and $\Omega$ is the vacuum vector.
The only dynamics that preserves the coherent structure is the one generated by the following Hamiltonian. Let $\omega$ be a self-adjoint operator on $\mathscr{H}$, and $e^{-it\omega}$ the associated one-particle evolution. Then the second quantizations $\mathrm{d}\Gamma(\omega)$ and $\Gamma(e^{-it\omega})=e^{-it\mathrm{d}\Gamma(\omega)}$ are defined by the action on each $n$-particle subspace $\mathscr{H}_n$: $$\bigl(\mathrm{d}\Gamma(\omega)\psi\bigr)_n= \bigl(\sum_{j=1}^n 1\otimes \dotsm\otimes 1\otimes\omega_j\otimes 1\otimes\dotsm\otimes 1\bigr)\psi_n\;,$$ where $\omega_j$ is the operator $\omega$ acting only on the $j$-th variable; $$\bigl(\Gamma(e^{-it\omega})\psi\bigr)_n=\bigl(\prod_{j=1}^n e^{-it\omega_j}\bigr)\psi_n\; .$$
Now, it is not difficult to prove that for all $\alpha\in\mathscr{H}$, $$\Gamma(e^{-it\omega})\lvert\alpha\rangle=\lvert\alpha_t\rangle=\lvert e^{-it\omega}\alpha\rangle\; .$$ Therefore, the coherent structure is preserved by a $\Gamma$ evolution. This is, however, not true for other evolutions! A $\Gamma$ evolution is usually what models a free theory, but not interacting ones.
In quantum mechanics, since $\mathscr{H}=\mathbb{C}$, $\omega$ can only be a real number, and $\mathrm{d}\Gamma(\omega)$ is thus a (rescaled) harmonic oscillator (in fact $\mathrm{d}\Gamma(1)$ is the number operator). Therefore, in quantum mechanics only the harmonic oscillator dynamics leaves the coherent structure invariant.