First-pass perfusion cardiac MR (FPP-CMR) allows the assessment of ischemic heart disease. However, conventional FPP-CMR has limited spatial resolution and requires breath-holding. Moreover, diagnostic accuracy may be compromised due to dark-rim artifacts. Here, we propose a free-breathing quantitative high-resolution FPP-CMR framework that combines dynamically variable undersampling with a motion-corrected reconstruction approach that uses spatial and temporal constraints. The proposed motion-corrected strategy improves image sharpness and quantification of myocardial blood flow from free-breathing acquisitions. The highly undersampled acquisitions allow short acquisition windows and facilitate higher spatial resolution images, which are less sensitive to dark-rim artifacts, thus improving the diagnostic accuracy of FPP-CMR.