This paper is concerned with the existence and multiplicity of nontrivial solutions for a class of Kirchhoff-type systems in $\mathbb{R}^N$ involving the fractional pseudo-differential operators defined as the generalizations of the $p(x)$-Laplace operator. Our main tools come from a direct variational methods, the Mountain Pass Theorem, the symmetric Mountain Pass Theorem and the Fountain Theorem in critical point theory. The obtained results of this note significantly contribute to the study of Kirchhoff-type systems in the sense that our situation covers not only differential operators of fractional order but also nonhomogeneous differential operators in Sobolev spaces with variable exponent.