We study the quench dynamics of entanglement spectra in the Kitaev chain with variable-range pairing quantified by power-law decay rate $alpha$. Considering the post-quench Hamiltonians with flat bands, we demonstrate that the presence of entanglement-spectrum crossings during its dynamics is able to characterize the topological phase transitions (TPTs) in both short-range ($alpha$ > 1) or long-range ($alpha$ < 1) sector. Properties of entanglement-spectrum dynamics are revealed for the quench protocols in the long-range sector or with $alpha$ as the quench parameter. In particular, when the lowest upper-half entanglement-spectrum value of the initial Hamiltonian is smaller than the final one, the TPTs can also be diagnosed by the difference between the lowest two upper-half entanglement-spectrum values if the halfway winding number is not equal to that of the initial Hamiltonian. Moreover, we discuss the stability of characterizing the TPTs via entanglement-spectrum crossings against energy dispersion in the long-range model.
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