The circumgalactic medium (CGM) is the outermost, gaseous envelope of a galaxy, spanning beyond the visible stellar disk and dominates the galaxy's baryonic mass. This expansive gas reservoir plays an influential role in cosmic structure formation and records critical information about a galaxy’s past and ongoing interactions with the surrounding environment. Understanding the detailed physical properties of the CGM is a vital step to improving the current galaxy evolution theories. In particular, evidence has been mounting that the ebbs and flows of baryonic matter in the CGM play a crucial role in driving galaxy formation, maturation, and eventual quiescence. While recent CGM surveys have significantly tightened constraints on the spatial extent and column density of the gas, our direct observations of gas motions remain notably limited. In this thesis, I leveraged the exquisite sensitivity of the latest generation of integral-field spectrographs to provide empirical constraints on two key dynamical processes in the CGM: galactic superwinds driven by star-forming galaxies and the subsonic turbulent energy cascade in the low-density halo environment. Taking advantage of the magnifying power of strong gravitational lensing and employing Lyman-alpha radiative transfer models, I recovered highly organized velocity fields across galactic outflow regions at scales of ≈5--30 kpc surrounding star-forming galaxies at z≈3--4 (Chapter 2). Utilizing two-point statistical analyses derived from the spatially-resolved kinematic measures, I uncovered the subsonic nature of gas motions at scales of ≈10--60 kpc in the quasar CGM at z≈0.5--1 (Chapters 3 and 4). These empirical results shed light on the intimate connection between galaxies and their surrounding CGM, illuminating the role of star-formation/AGN feedback and galaxy environments on the evolution of the baryon cycle over cosmic time.