Consider a stochastic process $\{X(t)\}$ on a finite state space $ {\sf X}=\{1,\dots, d\}$. It is conditionally Markov, given a real-valued `input process' $\{ζ(t)\}$. This is assumed to be small, which is modeled through the scaling, \[ ζ_t = \varepsilon ζ^1_t, \qquad 0\le \varepsilon \le 1\,, \] where $\{ζ^1(t)\}$ is a bounded stationary process. The following conclusions are obtained, subject to smoothness assumptions on the controlled transition matrix and a mixing condition on $\{ζ(t)\}$: (i) A stationary version of the process is constructed, that is coupled with a stationary version of the Markov chain $\{X^\bullet$(t)\}obtained with $\{ζ(t)\}\equiv 0$. The triple $(\{X(t)\}, \{X^\bullet(t)\},\{ζ(t)\})$ is a jointly stationary process satisfying \[ {\sf P}\{X(t) \neq X^\bullet(t)\} = O(\varepsilon) \] Moreover, a second-order Taylor-series approximation is obtained: \[ {\sf P}\{X(t) =i \} ={\sf P}\{X^\bullet(t) =i \} + \varepsilon^2 \varrho(i) + o(\varepsilon^2),\quad 1\le i\le d, \] with an explicit formula for the vector $\varrho\in\mathbb{R}^d$. (ii) For any $m\ge 1$ and any function $f\colon \{1,\dots,d\}\times \mathbb{R}\to\mathbb{R}^m$, the stationary stochastic process $Y(t) = f(X(t),ζ(t))$ has a power spectral density $\text{S}_f$ that admits a second order Taylor series expansion: A function $\text{S}^{(2)}_f\colon [-π,π] \to \mathbb{C}^{ m\times m}$ is constructed such that \[ \text{S}_f(θ) = \text{S}^\bullet_f(θ) + \varepsilon^2 \text{S}_f^{(2)}(θ) + o(\varepsilon^2),\quad θ\in [-π,π] . \] An explicit formula for the function $\text{S}_f^{(2)}$ is obtained, based in part on the bounds in (i). The results are illustrated using a version of the timing channel of Anantharam and Verdu.