The Wikipedia article about the Statute of Westminster 1931 pretty much answers this question:
The preamble to the Statute of Westminster sets out conventions which affect attempts to change the rules of succession to the Crown. The second paragraph of the preamble to the Statute reads:
[...] it would be in accord with the established constitutional position of all the members of the Commonwealth in relation to one another that any alteration in the law touching the Succession to the Throne or the Royal Style and Titles shall hereafter require the assent as well of the Parliaments of all the Dominions as of the Parliament of the United Kingdom:
This means, for example, that any change in any realm to the Act of Settlement's provisions barring Roman Catholics from the throne would require the unanimous assent of the parliaments of all the other Commonwealth realms if the shared aspect of the Crown is to be retained.
In other words, all Commonwealth Realms must have identical succession rules.
It also adds:
The preamble does not itself contain enforceable provisions, it merely expresses a constitutional convention, albeit one fundamental to the basis of the relationship between the Commonwealth realms. (As sovereign nations, each is free to withdraw from the arrangement, using their respective process for constitutional amendment.)
As for the most recent change to royal succession, the principles were agreed by all the Commonwealth Realms in the Perth Agreement in 2011. Those realms which required changes to their domestic legislation took the necessary steps, and the change took effect in all the realms on 26 March 2015.