Velocity-resolved reverberation mapping (RM) is a powerful technique to constrain the geometry and kinematics of the AGN broad-line region via measuring the delayed responses (lags) of the broad-line emission at various velocities to the driving continuum variations. Velocity-resolved lags, the inferred 2D (velocity and lag) transfer function of RM, and dynamical modeling of time-resolved spectroscopy provide unique and critical constraints on the BLR, enabling robust measurements of the black hole mass of the AGN. However, given the stringent sampling and SNR requirements, velocity-resolved RM has been largely limited to bright nearby AGN and to the broad Hbeta line. To explore the diversity of AGN broad-line region for other broad lines and for different accretion parameters (e.g., black hole mass and luminosity), it is necessary to expand the sample of velocity-resolved RM. Here we propose a Gemini Large and Long program to measure velocity-resolved RM for a sample of 12 broad-line AGN across a broad range of redshifts (0.2<z<2) and black hole masses (1E8-1E9 Msun), using multi-epoch GMOS spectroscopy to provide high-SNR measurements of reverberation signals in the wings of the broad lines. This is the first systematic study of velocity-resolved RM beyond the nearby universe and for broad MgII and CIV lines. Importantly, we combine the GMOS spectroscopy with dense photometric sampling and more than 100 spectroscopic epochs during 2023B-2026A from the ongoing SDSS-V Black Hole Mapper Reverberation Mapping program, and additional imaging and spectroscopic monitoring resources acquired by our team. The GMOS spectra provide the necessary measurements of flux responses in the broad-line wings, while the concurrent photometric and additional spectra from smaller-aperture facilities greatly reduces the demands on GMOS epochs, making it possible to accommodate a diverse sample with a reasonable amount of Gemini time. Being at the right time and with the right resources, this program is promising to deliver the first comprehensive view of the geometry and kinematics of the AGN broad-line region diversified in redshift, line species, Eddington ratio, and BH masses, and will provide an unprecedented data set to significantly advance the RM field and synergize with complementary probes of the broad-line region. This program will drive forward a better understanding of the inner structures of AGN, a necessary step towards accurate AGN black hole mass estimation across cosmic time.