Historic England has designated the site as a Scheduled Ancient Monument. Their designation listing can be found here and includes the reasons for the listing, a history of the site and summary of the archaeological investigations to date.
The first investigations were those of Norman Norris in the 1930s, digging to the northeast of the standing buildings. His research was halted by the outbreak of war, but his reports of his findings can be read here (1935/6) and here (1937/9). These are courtesy of The Suffolk Institute of Archaeology and History.
Stanley West dug on the line of Pales Dyke in 1970, his report can be read here, again courtesy of The Suffolk Institute of Archaeology and History.
Stuart Boulter, Suffolk County Council Archaeological Service, investigated the central building complex in 1999 and established, amongst other things, the location and scale of the friary church, and found a burial ground to the north of the buildings. Unfortunately, his report does not appear to be available online, but can be obtained through the County Council archive, see here. The reference is 99/21.
Stuart Boulter and his team returned to the site in 2008 following the collapse of a section of the west perimeter wall in October 2007. They examined the wall foundations to determine the construction methods. A detailed photographic record was made of the central ruins and perimeter, and used to establish the dates of construction. Again, this is available through the County Council archive as above, reference 2008/52.
Channel 4's Time Team programme visited Dunwich in 2011, and dug two trenches on the Greyfriars site. One confirmed the site of the friary church, the second was a cross section of Pales Dyke and postulated a Saxon construction. The archaeology report, by Wessex Archaeology, can be read here, and the broadcast programme can be found here.
Professor David Sear and his team from Southampton University in 2014 investigated Pales Dyke and the major roads into the medieval town at the points where they outcrop at the cliff. In the case of Pales Dyke carbon dating of the organic material at its base, which would have been surface at the time it was dug, produced a central date of 375BC, suggesting PD could have been of Iron Age construction. The full report can be accessed here.
Most recently, in 2015, Professor Carenza Lewis and a team from Cambridge, led a community dig, sinking three trenches between Pales Dyke and the cliff edge, that is on land that would have been within the medieval town. Aside from a good deal of Saxon and medieval pottery shards, one trench uncovered the floors of two 12th/13th century dwellings. The full report can be accessed here.