Suppose that $Y$ is a scalar and $X$ is a second-order stochastic process, where $Y$ and $X$ are conditionally independent given the random variables $ξ_1,...,ξ_p$ which belong to the closed span $L_X^2$ of $X$. This paper investigates a unified framework for the inverse regression dimension-reduction problem. It is found that the identification of $L_X^2$ with the reproducing kernel Hilbert space of $X$ provides a platform for a seamless extension from the finite- to infinite-dimensional settings. It also facilitates convenient computational algorithms that can be applied to a variety of models.