What are the proposed theoretical explanations for the EmDrive? The EmDrive is a proposed propulsion mechanism for spacecraft in which some form of microwave device provides the propulsion. The consensus from the physics community, including many voices on this site, is that the proposed device would violate conservation of momentum. 
Several experiments have been conducted, by three main groups: (i) Roger Shawyer, from Satellite Propulsion Research Ltd, (ii) a research group at the Northwestern Polytechnical University in Xi'an, China, and (iii) Eagleworks Laboratories at NASA's Lyndon B. Johnson Space Center in Houston, Texas. However, few of these experiments have been performed in vacuum (and none of the vacuum tests has been published in a peer-reviewed journal), which means the effect of air currents around the device cannot be ruled out.
This device has been the subject of many questions on this site, including


*

*What exactly is NASA's proposed mechanism for "propellantless" "EM Drive" propulsion?

*Is the EmDrive, or "Relativity Drive" possible?

*NASA's "Impossible" Space Engine

*Has NASA confirmed that Roger Shawyer's EmDrive thruster works?

*https://physics.stackexchange.com/questions/185380/emdrive-in-relation-to-newest-nasa-article-does-it-extract-energy-out-of-the-q?lq=1
and their many closed duplicates.
Many of these questions ask what the proposed explanations are, but then veer off into territory of the form "assuming it is true, how does it work?", so they have been (rightfully) closed as counterfactual physics. I feel this sort of bundling does a lot of harm to the discussion, and there is some scope to ask what exactly is being proposed as an explanation. So:

What do the proponents of EmDrive claim is the theoretical justification of the device's function?

Please limit answers exclusively to references, and indicate whether they are peer-reviewed or not. Please limit answers to serious attempts at explanations, as opposed to vague claims about the 'quantum vacuum' - or indicate if an experimental paper does not have any such theoretical explanation or reference.
To be clear, this is not the place to discuss whether the explanations are correct or not. If you have specific questions about any of these sources, ask it separately. (However, if your question starts with "assuming that this works", be prepared for it to be closed.) If you feel there are not enough resources on this thread, you can draw attention to it but please do not post another question asking people to look for more references.
 A: (i) Roger Shawyer
Shawyer's output seems to be mostly available on emdrive.com. Among the theoretical explanations he provides there are

*

*A Note on the Principles of EmDrive force measurement

*Principle of Operation

*Theory paper
None of these appear to be peer-reviewed.
(ii) NWPU group


*Applying Method of Reference 2 to Effectively Calculating Performance
of Microwave Radiation Thruster. Yang Juan, Yang Le, Zhu Yu, Ma Nan. Journal of Northwestern Polytechnical University 28 no. 6, 807 2010. English translation on emdrive.com.


*Net thrust measurement of propellantless microwave thruster. Yang Juan, Wang Yu-Quan, Li Peng-Fei, Wang Yang, Wang Yun-Min, Ma Yan-Jie. Acta Phys. Sin. 61 no. 11, 110301 (2012). (In Chinese; English translation on emdrive.com).


*Prediction and experimental measurement of the electromagnetic thrust generated by a microwave thruster system. Yang Juan, Wang Yu-Quan, Ma Yan-Jie and Li Peng-Fei. Chin. Phys. B 22 no. 5 050301 (2013).
It's important to note, also, that at least some of the NWPU results appear to have been retracted, with a correction published as J. Prop. Tech. 37 no. 2, p. 362 (2016) (doi, pdf, abstract in English and full text in Chinese).
(iii) Eagleworks Laboratory
Somewhat surprisingly, there is a peer-reviewed NASA publication on the matter,


*Anomalous Thrust Production from an RF Test Device Measured on a Low-Thrust Torsion Pendulum. David Brady, Harold White, Paul March, James Lawrence and Frank Davies. 50th AIAA / ASME / SAE / ASEE Joint Propulsion Conference, American Institute of Aeronautics and Astronautics, 2014.

The authors' affiliations are all listed as Eagleworks Laboratories, Lyndon B. Johnson Space Center, Houston, Texas. There is also a popular summary on NASA's technical reports server. This is a purely experimental paper, with no tests in vacuum. Their only references are to [6] and to cannae.com.
There is another related paper on that conference, presented by Guido P. Fetta of Cannae, LLC,


*Numerical and Experimental Results for a Novel Propulsion Technology Requiring no On-Board Propellant. Guido P. Fetta. 50th AIAA/ASME/SAE/ASEE Joint Propulsion Conference, American Institute of Aeronautics and Astronautics, 2014.

This is paywalled so it is difficult to evaluate.
On the NASA connection, it is important to emphasize that NASASpaceflight, a science and space outreach website which covers the EmDrive relatively often on site (e.g. this article) and in its forum, does not have any official relation to NASA. Questions claiming NASA support for results published or endorsed by NASASpaceflight, and which have not appeared on e.g. NASA's Technical Reports Server, indicate poor prior research and are much more likely to be closed on this site.
A more recent NASA paper has also appeared in the peer-reviewed literature, in


*Measurement of Impulsive Thrust from a Closed Radio-Frequency Cavity in Vacuum. H. White et al. J. Propul. Power, advance online eprint (2016).

This paper includes a discussion section (§II.C.10) describing some of the models the authors think can be used to describe the effect.
(iv) M.E. McCullooch at SMSE, Plymouth University


*Testing quantised inertia on the emdrive. M.E. McCulloch. Europhys. Lett. 111, 60005 (2015), arXiv:1604.03449. (Peer reviewed.)

(v) Lisbon group


*A possible explanation for the EM drive based on pilot wave theory. J.R. Croca, P. Castro, M. Gatta and L. Gurriana. J. Appl. Phys. Sci. Int. 8 no. 4, 145-151 (2017). Journal claims it is peer reviewed; on the other hand, the publisher, International Knowledge Press, appears in Beall's list, and several other hallmarks (such as the lack of a DOI) suggest only a cursory peer review at best.

