The purpose of this study is to posit a principled explanation for the cross-linguistic puzzling phenomena of V-v-to-T and T-to-C movements found in English and French. For the explanation, Duality of interpretive properties and Phase Head are operated for the Agree valuation within the phase instead of the features of [Weak] and [Strong], admitting the next phase evaluation of look-ahead. For the further operations of verb (V)-movements, Chomsky's (2013, 2014) minimal search labeling algorithm (LA) interpreted in Interface and Matushansky’s (2006) DM are adopted and slightly modified. Namely, headhead shared features as the label and m-merger for morphological processes are used in narrow syntax and interpreted in Interface. And Present Form as in -s, Past Form as in -ed, Modal Form such as can or do, Perfect Form have-ed, Progressive Form be-ing, and Passive Form be-ed are suggested to be merged in the v like causative make on the ground of slightly fixed Bare Approach. As a result, the Head Movement Constraint (HMC), do-support and affix-lowering are addressed without violating Extension Condition (EC) and c-command.