This work introduces a novel near real-time (real-time after an initial short delay) MRI solution that handles motion well and is generalizable. Here, real-time means the algorithm works well on a highly accelerated scan, is zero-latency (reconstructs a new frame as soon as MRI data for it arrives), and is fast enough, i.e., the time taken to process a frame is comparable to the scan time per frame or lesser. We demonstrate its generalizability through experiments on 6 prospective datasets and 17 retrospective datasets that span multiple different applications -- speech larynx imaging, brain, ungated cardiac perfusion, cardiac cine, cardiac OCMR, abdomen; sampling schemes -- Cartesian, pseudo-radial, radial, spiral; and sampling rates -- ranging from 6x to 4 radial lines per frame. Comparisons with a large number of existing real-time and batch methods, including unsupervised and supervised deep learning methods, show the power and speed of our approach.