Through comprehensive analysis of carboxylate-based metal-organic frameworks (MOFs), we present general evidence that challenges the common perception of MOF metal-linker bonds being static. Structural dynamics in MOFs, however, typically refers to the "breathing" behavior of cavities, where pores open and close in response to guest molecules, and to the transient binding of guest molecules, but dynamic bonding would explain important MOF phenomena in catalysis, postsynthetic exchange, negative thermal expansion, and crystal growth. Here, we demonstrate, through use of variable-temperature diffuse reflectance infrared Fourier transform spectroscopy (VT-DRIFTS) aided by ab initio plane wave density functional theory, that similar evidence for melting behavior in zeolitic imidazolate frameworks (ZIFs), i.e., reversible metal-linker bonding, driven by specific vibrational modes, can be observed for carboxylate MOFs by monitoring the red-shifts of carboxylate stretches coupled to anharmonic metal-carboxylate oscillators. To demonstrate the generality of these findings, we investigate a wide class of carboxylate MOFs that includes iconic examples with diverse structures and metal-linker chemistry. As the very vibrations invoked in ZIF melting but heretofore unobserved for carboxylate MOFs, these metal-linker dynamics resemble the ubiquitous soft modes that trigger important phase transitions in diverse classes of materials while offering a fundamentally new perspective for the design of next-generation metal-organic materials.