Covid mRNA vaccines are injected into the deltoid. What is the process in which they spread from there to the rest of the body? Would there be a better immunization reaction if the second dose is injected elsewhere?
The short answer to how vaccines are spread around the body is, quickly.
What is the process in which they spread from there to the rest of the body?
The reason vaccines are often delivered into muscle tissue is because muscle tissue is highly vascularized with both blood and lymphatic vessels. Immediately after a vaccine injection, the delivery vehicle (probably lipid nanoparticles or a viral vector) will quickly start diffusing into the blood, where it will be carried up to the heart, and from there begin spreading around the body (a lot of it ends up in the liver). Somewhat less quickly, in the lymph it will essentially start to generate an immune response immediately as it drains into lymph nodes and begins circulating throughout the lymphatic system (note the the desired immune response won't start until the genetic component of a vaccine has entered nucleated cells and has started being translated into the desired viral antigens).
Would there be a better immunization reaction if the second dose is injected elsewhere?
Don't know if I can definitively justify saying "no" to this question, but probably not. Muscle tissue seems to be the best injection site for vaccines. Deltoids are probably the best muscle because there's a lower chance of missing the intended injection site compared with some other muscles (e.g. glutes). And the systemic spread of vaccine components starts happening pretty fast after injection. In one animal study, researchers surgically removed a leg muscle 1 to 10 minutes after injection and still didn't prevent the related antibody response, or even really reduce it compared to animals with the muscle left intact.