Surface acoustic wave (SAW) tweezers are a promising multifunctional micromanipulation method that controls microscale targets via patterned acoustic fields. Owing to their device structure and bonding process, most SAW tweezers have limitations in terms of controlling the position and motion of the acoustic traps, as they generate an acoustic field with a fixed region and adjust the manipulation effects via signal modulation. To address this challenge, we propose movable SAW tweezers with a multilayer structure, achieving dynamic control of their wave field and acoustic trap positions; we demonstrate their precise manipulation functions, such as translation, in-plane rotation, out-of-plane rotation, and cluster formation, on a wide spectrum of samples, including particles, bubbles, droplets, cells, and microorganisms. Our method not only improves the degree of freedom and working range of SAW tweezers but also allows for precise and selective manipulation of microtargets via microtools and localized wavefields. Owing to their flexibility, versatility, and biocompatibility, the movable SAW tweezers can be a practical platform for achieving arbitrary manipulation of microscale targets and have the potential to play significant roles in biomedical microrobotics.