I’m a little bit vague on some of the detail, but as I understand it the position is as follows.
Succession to the British crown is regulated by the Act of Settlement, 1701. The crown passes to the heir of Sophie, Electress of Hanover, provided that person is a Protestant.
Canada, Australia, etc, started out as colonies, politically and constitutionally completely dependent on the British crown. The British monarch, in his or her capacity as the British monarch, was at the head of the governmental hierarchy. There was no question of separate crowns being held by the same individual (as was the case with English and Scottish crowns from 1603 to 1707, and with the English/British and Irish crowns from 1536 to 1801).
In due course the larger (and, initially, the whiter) colonies moved towards self-government. Canada was granted “dominion status” in 1867, Australia in 1901, New Zealand in 1907, and so forth. Initially this meant, in effect, internal representative self-government. Canadian electors voted for a Canadian parliament from which a Canadian prime minister was appointed and the monarch (represented by the Governor-General) acted on the advice of Canadian ministers in matter relating exclusively to Canada. There was no question, however, that a separate crown had been created; just a separate set of advisers to advise the undivided British crown in relation to matters exclusively affecting one particular territory.
Over time, dominion status was enlarged. The governor-general ceased to be chosen by the Westminster government, and instead was chosen by the dominion government. Dominion governments played an increasing role in foreign affairs, initially with their immediate neighbours but in time with states further and further away. Dominions armed forces grew larger and more diverse, and they relied less and less on British forces for defence, and so on.
Their legal powers were enlarged also. In (I think) 1927 each dominion was given power (by British legislation) to enact its own legislation conferring titles on the crown for use within the dominion. And the (British) Statute of Westminster 1931 allowed a dominion to pass legislation altering existing British law in force within that dominion, and forbade the British parliament to make laws having force within a dominion except at the request of that dominion. This is generally regarded as marking the formal legal and political independence of the dominions. Certain constitutional legislation was ring-fenced and the Statute of Westminster did not apply to it, but the various ring-fencing provisions have been progressively removed since then.
It is now open to the various dominions to enact constitutional legislation creating a separate crown to be held by the person who holds the British crown (assuming that person agrees, of course) but none of them have done so. Those former dominions which no longer wish to be constitutionally to the British crown have moved to republican status, e.g. India, Ireland, South Africa.
If the British parliament were now to enact legislation altering the Act of Settlement (e.g. by providing for parity as between men and women in the succession to the crown, or removing the requirement that the monarch be a Protestant, both of which have been mooted) that legislation would not apply in any dominion, unless that dominion requested and accepted it. Indeed, that has been used as a reason to avoid altering the succession; most of the dominons have more pressing issues to worry about and, if the British government were to raise this as an issue requiring attention, the forseeable result would be to reinforce in a number of dominions the idea that the link to the British crown is, for them, a distraction and an irrelevance, and to impel them towards republicanism.
The term “dominion” is not much used nowadays but, when it is, it describes those member states of the British Commonwealth which have the British monarch as their head of state. There are some members states which have a separate crown with a different monarch (e.g. Tonga); they are not regarded as dominions. Quite how a state would be classified if it had a separate crown occupied by the same person as the British crown is not clear, but the point is not likely to arise; there is no logic to breaking links with the British Crown but retaining Elizabeth of Windsor as head of state.