Are the qualifications the same

No, the educational requirements are not the same. If you came to the US with a certification in early childhood education, or certification in child care, you would need additional training to meet requirements - and each state has their own requirements.

Same goes for someone who leaves the US with credentials and goes to another country. The training received may well not meet criteria. I only know this because one of my co-workers moved to Queensland and her qualifications did not meet what she needed.

Just do some homework and see what is needed for where you were living and match that with what you you want to continue while in the US.