Approaches to the Internalism-Externalism controversy in the philosophy of mind often involve both (broadly) metaphysical and explanatory considerations. Whereas originally most emphasis seems to have been placed on metaphysical concerns, recently the explanation angle is getting more attention. Explanatory considerations promise to offer more neutral grounds for cognitive systems demarcation than (broadly) metaphysical ones. However, it has been argued that explanation-based approaches are incapable of determining the plausibility of internalist-based conceptions of cognition vis-a-vis externalist ones. On this perspective, improved metaphysics is the route along which to solve the Internalist-Externalist stalemate. In this paper we challenge this claim. Although we agree that explanation-orientated approaches have indeed so far failed to deliver solid means for cognitive system demarcation, we elaborate a more promising explanation-oriented framework to address this issue. We argue that the mutual manipulability account of constitutive relevance in mechanisms, extended with the criterion of 'fat-handedness', is capable of plausibly addressing the cognitive systems demarcation problem, and thus able to decide on the explanatory traction of Internalist vs. Externalist conceptions, on a case-by-case basis. Our analysis also highlights why some other recent mechanistic takes on the problem of cognitive systems demarcation have been unsuccessful. We illustrate our claims with a case on gestures and learning. (C) 2016 Elsevier Ltd. All rights reserved.