Kinesin-14s, a subfamily of the large superfamily of kinesin motor proteins, function mainly in spindle assembly and maintenance during mitosis and meiosis. KlpA from Aspergillus nidulans and GiKIN14a from Giardia intestinalis are two types of kinesin-14s. Available experimental results puzzlingly showed that while KlpA moves preferentially toward the minus end in microtubule-gliding setups and inside parallel microtubule overlaps, it moves preferentially toward the plus end on single microtubules. More puzzlingly, the insertion of an extra polypeptide linker in the central region of the neck stalk switches the motility direction of KlpA on single microtubules to the minus end. Prior experimental results showed that GiKIN14a moves preferentially toward the minus end on single microtubules in either tailless or full-length forms. The tail not only greatly enhances the processivity but also accelerates the ATPase rate and velocity of GiKIN14a. The insertion of an extra polypeptide linker in the central region of the neck stalk reduces the ATPase rate of GiKIN14a. However, the underlying mechanism of these puzzling dynamical features for KlpA and GiKIN14a is unclear. Here, to understand this mechanism, the dynamics of KlpA and GiKIN14a were studied theoretically on the basis of the proposed model, incorporating potential changes between the kinesin head and microtubule, as well as the potential between the tail and microtubule. The theoretical results quantitatively explain the available experimental results and provide predicted results. It was found that the elasticity of the neck stalk determines the directionality of KlpA on single microtubules and affects the ATPase rate and velocity of GiKIN14a on single microtubules.