How do ASKAP's focal plane phased array feeds interact with the entire array phasing?
The phased array feeds (PAFs) on ASKAP are indeed located at the focal plane. The ASKAP PAFs have 188 receivers, and from this digital beam-forming will create 36 independent beams on the sky. Some more information can be found from the CSIRO. Imaging is then done by scanning the dish (or letting the sky roll overhead). The scanning can then be done much faster than with a single receiver.

Its important not to confuse these independent beams with purpose of the array itself. The main reason for using an array is interferometry. We use an array for this purpose to have multiple baselines (the distance between the antennas). Having multiple baselines allows the telescope to 'see' structure various angular resolutions. It's true that you can use the array in a 'fly's eye' mode, and then every dish is providing 36 independent beams. In this mode even larger areas of the sky can be scanned quickly. This is useful for searching for transient events such as fast radio bursts.

So, the phasing within the PAF needs to be correlated and the beams formed digitally, and then the signals from each dish need to correlated. A huge task indeed! The computing breakdown for ASKAP is given here. A new supercomputer (Pawsey) has been built in WA for the purpose of processing ASKAP data.