The article Tectonic setting of kimberlites (Jelsma et al. 2009) has made correlations between older kimberlite volcanics and aspects of the supercontinent cycle, specifically:
Kimberlites can be viewed as time capsules in a global plate tectonic framework. Their distribution illustrates clustering in time and space. Kimberlite ages span the assembly and break-up of a number of supercontinents, such as Rodinia and Gondwana. These supercontinents show time lines with (i) broad periods devoid of kimberlite magmatism corresponding to times of continent stability, and (ii) narrow kimberlite emplacement windows corresponding to times of fundamental plate reorganizations.
Implying a link between parts of the supercontinent cycle and kimberlite magmatism.
However, this implied tectonic setting does not seem to match with Quaternary Tanzania, when the Igwisi Hills kimberlite volcanoes erupted. Therefore, what are the tectonic implications of the Quaternary Igwisi Hills, Tanzania, kimberlite volcanics?
The only thing that comes to mind is the East African Rift tectonics, but are not sure about this link.
Seismic studies and observations detected that the cratonic lithosphere base is approximately 170km, after which, there is anomalously low velocities to a depth of about 350km (1), which according to models described by Weeraratne et al. (1) indicate that beneath the cratonic crust, there is the presence of high temperature and the presence of melt:
The mineralogy of xenoliths in the pyroclasts and lava further indicate that there has been small degrees of carbonate-apatite bearing peridotite source melts from depths greater than 110km. (2)
Brown et al. (3) conclude that the fact that these kimberlitic volcanics are so comparatively young, they may indicate that a new phase of kimberlite activity may be starting in the Tanzanaian craton, especially when considering the mantle plume and active rifting in the adjacent East Africa Rift branches. This suggests that a link between kimberlite volcanism and the supercontinent cycle maybe too much of a simplistic view, when the reality is probably closer to individual plate reorganisation (in this case, the eventual splitting of the African continent).
References
(1) Weeraratne et al. 2003, Evidence for an upper mantle plume beneath the Tanzanian craton from Rayleigh wave tomography, Journal and Geophysical Research.
(2) Dawson, 1994, Quaternary kimberlitic volcanism on the Tanzania Craton, Contributions to Mineralogy and Petrology (abstract only - paywalled)
(3) Brown et al. 2012, Eruption of kimberlite magmas: physical volcanology, geomorphology and age of the youngest kimberlitic volcanoes known on earth (the Upper Pleistocene/Holocene Igwisi Hills volcanoes, Tanzania), Explore Bristol Research (abstract only - paywalled)