I've been following this string with interest. Earlier this year, at the Bertinoro conference, I gave a paper on Behavioural Process, in which I argued that if one tries to synthesize what is said about Behavioural Process in various introductions, the result is contradictory and incoherent. At the risk of being called a heretic (again!), I suggest that it is preferable to uses a system with 5 process types, eliminating Behavioural Process. I treat Verbal Process as being processes of communication, within which it is possible to distinguish two sub-groups: those that project, and those that don't.
 Indeed, the treatment of behavioural processes in Deploying Functional Grammar (Martin, Matthiessen & Painter 2010) is clearly inconsistent with Halliday & Matthiessen (2004). For example, the former analyses the clause the tyres went 'screech!' as behavioural, whereas the latter characterises behavioural processes as processes of psychological and physiological behaviour. See 'Deploying Functional Grammar' On "Behavioural" Processes.
 Halliday & Matthiessen (2004: 248-50, 255) already acknowledge that:
They are the least distinct of all the six process types because they have no clearly defined characteristics of their own; rather they are partly like the material and partly like the mental. … ‘behavioural’ process clauses are not so much a distinct type of process, but rather a cluster of small subtypes blending the material and the mental into a continuum …
and suggest (Halliday & Matthiessen 1999: 136):
These can be interpreted as a subtype of material processes or as a borderline category between material and mental.
 Halliday & Matthiessen (2004: 256) already specify verbal processes that don't — or rarely — project reported locutions:
Verbs that accept a Target do not easily project reported speech; this type of clause is closer to the Actor + Goal structure of a ‘material’ clause …