Here's some research that you could also find online.
This is not an exhaustive list, obviously. Just meant as a jumping off place.
I'll likely keep adding research as I come across it, and as I have time.
If you are looking for research PubMed and Google Scholar are good places to start.
Akkoca Ö, Oğuz H, Ünlü CE, Aydın E, Ozdel K, Kavuzlu A. Association Between Nasal Obstruction Symptoms and Anxiety. Ear Nose Throat J. 2020 Aug;99(7):448-452. doi: 10.1177/0145561319900747. Epub 2020 Feb 13. PMID: 32050802.
Chambi-Rocha A, Cabrera-Domínguez ME, Domínguez-Reyes A. Breathing mode influence on craniofacial development and head posture. J Pediatr (Rio J). 2018 Mar-Apr;94(2):123-130. doi: 10.1016/j.jped.2017.05.007. Epub 2017 Aug 14. PMID: 28818510.
Courtney, R., et al. (2022). Functional nasal breathing rehabilitation: Effectiveness and feasibility of an online integrative breathing therapy protocol. International Journal of Orofacial Myology and Myofunctional Therapy, 48(1), 1-14.
Eby GA. Strong humming for one hour daily to terminate chronic rhinosinusitis in four days: a case report and hypothesis for action by stimulation of endogenous nasal nitric oxide production. Med Hypotheses. 2006;66(4):851-4. doi: 10.1016/j.mehy.2005.11.035. Epub 2006 Jan 10. PMID: 16406689.
Moris, Jose M., Arturo Cardona, Brendan Hinckley, Armando Mendez, Alexandra Blades, Vineet K. Paidisetty, Christian J. Chang, Ryan Curtis, Kylie Allen, Yunsuk Koh, A framework of transient hypercapnia to achieve an increased cerebral blood flow induced by nasal breathing during aerobic exercise, Cerebral Circulation - Cognition and Behavior, Volume 5, 2023.
Natasha Mirza, Donald C. Lanza, THE NASAL AIRWAY AND OBSTRUCTED BREATHING DURING SLEEP, Otolaryngologic Clinics of North America, Volume 32, Issue 2, 1999, Pages 243-262, ISSN 0030-6665, https://doi.org/10.1016/S0030-6665(05)70128-6.
This article refuting Polyvagal Theory by Grossman will be available here until July 26 (which is as long as Paul Grossman has said it will be free online). I've loaded it here as some have said they're having trouble getting the link.
In an effort to be thorough and transparent, here is Stephen Porges's response to critique.