The claims, like "meat, dairy, and eggs trigger an inflammatory reaction inside the body" arise from the studies in which they observed increased blood levels of "inflammatory markers," like C-reactive protein (CRP), in individuals on different diets. For example, in one study Dietary Red and Processed Meat Intake and Markers of Adiposity and Inflammation (J Am Coll Nutr, 2015) they observed an association between the high intake of red meat and increased CRP levels in women, but this does not already mean an association with any actual inflammatory disease.
In another study Vegetarian-Based Dietary Patterns and their Relation with Inflammatory and Immune Biomarkers: A Systematic Review and Meta-Analysis (Advances in Nutrition, 2019) they found an association between a vegetarian diet and reduced levels of inflammatory markers:
This study provides evidence that vegetarian-based dietary patterns
are associated with lowered serum C-reactive protein, fibrinogen, and
total leukocyte concentrations.
It could take years before high consumption of animal protein would result in an inflammatory disease, but it is not possible to make a study in which one group of participants would eat only animal and the other group only plant protein (and nothing else) for several years, so currently it is not possible to show a clear cause-effect relationship between animal protein and chronic inflammation.
Further, if a study says that a certain disease is associated with high animal protein consumption, it could be also associated with high animal fat or heme iron (from meat and fish) consumption, for example.
Examples of diseases with chronic inflammation:
1) Cardiovascular disease:
A review: Plant Protein and Animal Proteins: Do They Differentially Affect Cardiovascular Disease Risk? (Advances in Nutrition, 2015):
...evidence to date is inconclusive and is likely to remain so,
because it is difficult to isolate the independent effects of specific
Vegetarian, vegan diets and multiple health outcomes: A systematic review with meta-analysis of observational studies (CRSFN, 2017):
This comprehensive meta-analysis reports a significant protective
effect of a vegetarian diet versus the incidence and/or mortality from
ischemic heart disease (-25%) and incidence from total cancer (-8%).
Vegan diet conferred a significant reduced risk (-15%) of incidence
from total cancer.
3) Rheumatoid arthritis
Nutrition Interventions in Rheumatoid Arthritis: The Potential Use of Plant-Based Diets. A Review (Frontiers in Nutrition, 2019):
Excessive body weight and diets that include animal products (e.g.,
dairy, red meat) exacerbate the RA symptoms likely due to their
pro-inflammatory effects. In contrast, diets rich in vegetables,
fruits, and fiber are associated with lower BMI, have
anti-inflammatory properties and help reduce pain and inflammation in
4) Inflammatory bowel disease:
A prospective study including 67,581 women aged 40-65 years, in France: Animal protein intake and risk of inflammatory bowel disease (Am J Gastroenterol, 2010):
High total protein intake, specifically animal protein, was associated
with a significantly increased risk of IBD [inflammatory bowel disease], (hazards ratio for the
third vs. first tertile and 95% confidence interval being 3.31 and
1.41-7.77, and 3.03 and 1.45-6.34 for total and animal protein, respectively). Among sources of animal protein, high consumption of
meat or fish but not of eggs or dairy products was associated with IBD
In conclusion, many studies have found an association between animal protein and inflammation. None of these studies have proven that the protein itself causes inflammatory diseases, though. From animal foods, the consumption of eggs (a review one, two) and dairy (a review one, two, three, four) has usually not been associated with inflammatory diseases, but the consumption of meat, especially processed meat* (a review one, two) has been.
*What is processed meat? According to World Health Organization:
Processed meat refers to meat that has been transformed through
salting, curing, fermentation, smoking, or other processes to enhance
flavor or improve preservation. Most processed meats contain pork or
beef, but processed meats may also contain other red meats, poultry,
offal, or meat by-products such as blood. Examples of processed meat
include hot dogs (frankfurters), ham, sausages, corned beef, and
biltong or beef jerky as well as canned meat and meat-based
preparations and sauces.
In this context, the term meat processing does not include freezing and meat preparation directly before use (boiling, frying, salting, adding spices, etc.).
In individual studies they may not specifically define processed meat, but they refer to meats that contain substances that have been either added or formed during food processing (phosphates, nitrates, nitrites, -nitroso-compounds, polycyclic aromatic hydrocarbons, etc.). This is to show that the observed effects of high consumption of processed meat are not necessary due to protein but possibly due to added substances.