<p>In this paper, we study a class of nonlocal Kirchhoff-type boundary value problems driven by a generalized <i>p</i>(<i>x</i>)-Laplacian operator. The main feature of the problem lies in the presence of a nonlocal Kirchhoff term combined with variable exponent growth conditions, which leads to significant analytical difficulties within the associated variational framework. By employing critical point theory together with variational methods in variable exponent Sobolev spaces, and working under the Cerami compactness condition, we establish the existence and multiplicity of weak solutions. More precisely, we first prove the existence of at least one non-trivial weak solution. We then obtain a multiplicity result guaranteeing at least two distinct weak solutions. Finally, using a Fountain-type argument, we establish the existence of infinitely many weak solutions with unbounded energy levels.</p>