Background: Understanding the nature and extent of sensorimotor decline in aging individuals and those with neurocognitive disorders (NCD), such as Alzheimer’s disease, is essential for designing effective music-based interventions. Our understanding of rhythmic functions remains incomplete, particularly in how aging and NCD affect sensorimotor synchronization and adaptation to tempo changes. Objective: This study aimed to investigate how aging and NCD severity impact tapping to metronomes and music, with and without tempo changes. Methods: Patients from a memory clinic participated in a tapping task, synchronizing with metronomic and musical sequences, some of which contained sudden tempo changes. After exclusions, 51 patients were included in the final analysis. Results: Participants’ Mini-Mental State Examination scores were associated with tapping consistency. Additionally, age negatively influenced consistency when synchronizing with a musical beat, whereas consistency remained stable across age when tapping with a metronome. Conclusions: The results indicate that the initial decline of attention and working memory with age may impact perception and synchronization to a musical beat, whereas progressive NCD-related cognitive decline results in more widespread sensorimotor decline, affecting tapping irrespective of audio type. These findings underline the importance of customizing rhythm-based interventions to the needs of older adults and individuals with NCD, taking into consideration their cognitive as well as their rhythmic aptitudes.