Did the Messenger of Allah, may Allah's prayers and peace be upon him, forbid moving between mosques for Tarawih prayer, and command adherence to a single nearby mosque throughout Ramadan?

There is no objection to moving to a distant mosque for Tarawih prayers. All mosques are houses of Allah, and there is no text that commands adherence to the nearest mosque. This is permissible if the intention is to aid in humility and presence of heart, or to benefit from the Imam's recitation, or to increase one's steps and reward.