Recent years have witnessed the development of droplet-based microfluidics as a useful and effective tool for high-throughput analysis in biological, chemical and environmental sciences. Despite the flourishing development of droplet manipulation techniques, only a few methods allow for label-free and quantitative inspection of flowing droplets in microchannels in real-time and in three dimensions (3-D). In this work, we propose and demonstrate the application of a real-time quantitative phase microscopy (RT-QPM) technique for 3-D visualization of droplets, and also for full-field and label-free measurement of analyte concentration distribution in the droplets. The phase imaging system consists of a linear-CCD-based holographic microscopy configuration and an optofluidic phase-shifting element, which can be used for retrieving quantitative phase maps of flowing objects in the microchannels with a temporal resolution only limited to the frame rate of the CCD camera. To demonstrate the capabilities of the proposed imaging technique, we have experimentally validated the 3-D image reconstruction of the droplets generated in squeezing and dripping regimes and quantitatively investigated the volumetric and morphological variation of droplets as well as droplet parameters related to the depth direction under different flow conditions. We also demonstrated the feasibility of using this technique, as a refractive index sensor, for in-line quantitative measurement of carbamide analyte concentration within the flowing droplets.