We present analysis and results from a work in press in the Astrophysical Journal (Banzatti et al. 2020). We performed a synergic study of protoplanetary disks to investigate links between inner disk gas molecules and the large-scale migration of solid pebbles. The sample includes 63 disks where two types of measurements are available: i) spatially-resolved disk images revealing the radial distribution of disk pebbles (mm-cm dust grains), from millimeter observations with ALMA or the SMA, and ii) infrared molecular emission spectra as observed with Spitzer. The line flux ratios of H2O with HCN, C2H2, and CO2 all anti-correlate with the dust disk radius Rdust, expanding previous results found by Najita et al. (2013) for HCN/H2O and the dust disk mass. By normalization with the dependence on accretion luminosity common to all molecules, only the H2O luminosity maintains a detectable anti-correlation with disk radius, suggesting that the strongest underlying relation is between H2O and Rdust. If Rdust is set by large-scale pebble drift, and if molecular luminosities trace the elemental budgets of inner disk warm gas, these results can be naturally explained with scenarios where the inner disk chemistry is fed by sublimation of oxygen-rich icy pebbles migrating inward from the outer disk. Anti-correlations are also detected between all molecular luminosities and the infrared index n13−30, which is sensitive to the presence and size of an inner disk dust cavity. Overall, these relations suggest a physical interconnection between dust and gas evolution both locally and across disk scales. We discuss fundamental predictions to test this interpretation and study the interplay between pebble drift, inner disk depletion, and the chemistry of planet-forming material.