The transfer matrix method remains a widely used tool for the computation of eigensolutions in discrete and continuous dynamic systems, notably in structural and rotor dynamics. This paper presents a sensitivity and continuation analysis of eigensolutions obtained through the transfer matrix framework. Sensitivity analysis is conducted by evaluating the parametric derivatives of eigenvalues and eigenvectors directly from the system characteristic equation, providing insight into how small changes in model parameters influence system dynamics. Continuation techniques are applied to track the evolution of eigensolutions as parameters vary, enabling the identification of phenomena such as eigenvalue veering and crossings. Particular attention is given to numerical issues inherent to the transfer matrix method, and practical strategies are discussed to maintain solution accuracy during parameter continuation. The proposed approach is validated through representative examples of increasing complexity, demonstrating its effectiveness in capturing key trends and critical behaviors without requiring reformulation of the transfer matrix structure. The results suggest that sensitivity and continuation analyses offer a practical extension of the transfer matrix method, enhancing its application to parametric studies and preliminary design evaluations of complex dynamic systems.