The dynamics of probability density functions have been extensively studied in computational science and engineering to understand physical phenomena and facilitate algorithmic design. Of particular interest are dynamics formulated as gradient flows of energy functionals under the Wasserstein metric. The development of functional inequalities, such as the log-Sobolev inequality, plays a pivotal role in analyzing the convergence of these dynamics. This paper aims to extend the success of functional inequality techniques to dynamics that are gradient flows under the Fisher-Rao metric, with various $f$-divergences serving as energy functionals. Such dynamics take the form of nonlocal differential equations, for which existing analyses critically rely on explicit solution formulas in special cases. We provide a comprehensive study of functional inequalities and the relevant geodesic convexity for Fisher-Rao gradient flows under minimal assumptions. A notable feature of our functional inequalities is their independence from the log-concavity or log-Sobolev constants of the target distribution. Consequently, the convergence rate of the dynamics (assuming well-posedness) remains uniform across general target distributions.