Yes, the chest will move very slightly in deep belly breathing. But no, this is not a trivial thing, the focus in breathing should be on the abdomen/belly, which is your center; not the chest, which is a source of tension, and especially full of tension if you're breathing up "into" your chest.
Again, this is not a trivial thing, and you can easily check good sources on breathing (e.g., for meditation, singing, running) that will describe how belly breathing works. Chest breathing is actually not proper breathing in any circumstance; it doesn't use the diaphragm (the muscle designed for breathing purposes); it creates tension and doesn't allow breathing in as much air as belly-breathing; it's something we develop in our too-fast, too-stressful, too-nervous society.
These are just two top results that turned up in search for 'chest breathing' and meditation, you could find many, many more:
As a related note, breathing is of paramount importance in singing, which like meditation requires proper relaxation and deep breathing. Learning to how to belly breathe and get rid of bad habit of chest-breathing is one of the first things you'll learn from any singing instructor.
I have learned to control anxiety, and even fear, by simply pushing it from the chest down into the belly where it gets consumed. This was the phrasing someone used that I picked it up from, and it feels just like that. What I believe is really happening is really a realignment of breathing, just like you describe. It works.
Again, this is not a trivial thing, and you can easily check good sources on breathing (e.g., for meditation, singing, running) that will describe how belly breathing works. Chest breathing is actually not proper breathing in any circumstance; it doesn't use the diaphragm (the muscle designed for breathing purposes); it creates tension and doesn't allow breathing in as much air as belly-breathing; it's something we develop in our too-fast, too-stressful, too-nervous society.
These are just two top results that turned up in search for 'chest breathing' and meditation, you could find many, many more:
http://www.mayoclinic.org/tests-procedures/meditation/in-dep...
https://yogainternational.com/article/view/diaphragmatic-bre...
https://www.blinn.edu/counseling/Relaxation-Techniques.pdf
As a related note, breathing is of paramount importance in singing, which like meditation requires proper relaxation and deep breathing. Learning to how to belly breathe and get rid of bad habit of chest-breathing is one of the first things you'll learn from any singing instructor.