We propose a new recording modality, named here Space-Time Scanning Interferometry (STSI), which allows to capture interferograms using a linear array detector instead of a common 2D sensor. Object scanning is exploited to perform a different mapping of the holograms in the space-time domain. Three sensor rows are sufficient to yield the whole complex object field from a time sequence of interferograms. This approach is particularly useful in microfluidic microscopy, where the sample motion is intrinsically provided. We then introduce the Space-Time Digital Hologram (STDH), still possessing all the capabilities of a common DH, namely quantitative phase-contrast mapping of the samples and flexible refocusing starting from blind out-of focus recordings, but obtainable using a compact single line detector easily embeddable onboard a Lab-on-a-Chip platform. Above all, the proposed optofluidic approach is able to provide STDHs with unlimited FoV along the flow direction, independently of the set magnification factor and without the need for further processing such as hologram stitching. Hence, thanks to the possibility to refocus multiple flowing objects displaced in a liquid volume, STDH assures drastically enhanced throughput, quantitative and label-free, on-chip microscopy.