Aims. A census of molecular hydrogen flows across the entire Orion A giant molecular cloud is sought. With this paper we aim to associate each flow with its progenitor and associated molecular core, so that the characteristics of the outflows and outflow sources can be established. Methods. We present wide-field near-infrared images of Orion A, obtained with the Wide Field Camera, WFCAM, on the United Kingdom Infrared Telescope. Broad-band K and narrow-band H2 1-0S(1) images of a contiguous ~8 square degree region are compared to mid-IR photometry from the Spitzer Space Telescope and (sub)millimetre dust-continuum maps obtained with the MAMBO and SCUBA bolometer arrays. Using previously-published H2 images, we also measured proper motions for H2 features in 33 outflows, and use these data to help associate flows with existing sources and/or dust cores. Results. Together these data give a detailed picture of dynamical star formation across this extensive region. We increase the number of known H2 outflows to 116. A total of 111 H2 flows were observed with Spitzer; outflow sources are identified for 72 of them (12 more H2 flows have tentative progenitors). The MAMBO 1200 $\mu$m?maps cover 97 H2 flows; 57 of them (59%) are associated with Spitzer sources and either dust cores or extended 1200 $\mu$m?emission. The H2 jets are widely distributed and randomly orientated. The jets do not appear to be orthogonal to large-scale filaments or even to the small-scale cores associated with the outflow sources (at least when traced with the 11´´ resolution of the 1200 $\mu$m?MAMBO observations). Moreover, H2 jet lengths (L) and opening angles ($\theta$) are not obviously correlated with indicators of outflow source age – source spectral index, $\alpha$ (measured from mid-IR photometry), or (sub)millimetre core flux. It seems clear that excitation requirements limit the usefulness of H2 as a tracer of L and $\theta$ (though jet position angles are well defined). Conclusions. We demonstrate that H2 jet sources are predominantly protostellar sources with flat or positive mid-IR spectral indices, rather than disc-excess (or T Tauri) stars. Most protostars associated with molecular cores drive H2 outflows; however, not all molecular cores are associated with protostars or H2 jets. On statistical grounds, the H2 jet phase may be marginally shorter than the protostellar phase, though it must be considerably (by an order of magnitude) shorter than the prestellar phase. In terms of range and mean value of $\alpha$, H2 jet sources are indistinguishable from protostars. The spread in $\alpha$ observed for both protostars and H2 outflow sources is probably a function of inclination angle as much as source age. The few true protostars without H2 jets are almost certainly more evolved than their H2-jet-driving counterparts, although these later stages of protostellar evolution (as the source transitions to being a “disc-excess” source) must be very brief, since a large fraction of protostars do drive H2 flows. We also find that the protostars that power molecular outflows are no more (nor no less) clustered than protostars that do not. This suggests that the H2 emission regions in jets and outflows from young stars weaken and fade very quickly, before the source evolves from protostar to pre-main-sequence star, and on time-scales much shorter than those associated with the T Tauri phase, the Herbig-Haro jet phase, and the dispersal of young stellar objects.