Short answer is no it is not province-restricted and you can claim federal and provincial tuition and education (and textbook) amounts against tax payable on any taxable income, regardless of what province you earned it in.
N.B.: You should have earned both federal and provincial tuition credits in each year that you were in school. The crux is how the provincial credits are handled when moving. When moving from BC to Ontario, the form Schedule ON(S11) states that you can carry over your provincial credits as is. (That is, unless you're moving from Quebec, in which case your unused provincial amount must be set to your unused federal amount.) A related example would be moving to Alberta, where Schedule AB(S11) states that your unused provincial amount should be set to your unused federal amount no matter what. This rule may change from year to year as well.
So as long as you kept track of your provincial amounts when moving, you should be square.