First off, none of this is intended to be advice and is only meant to point you in the direction of the official sources.
For Australian permanent residency, it will depend on the visa you're after. For example take a look at the official eligibility criteria for the 189 visa
It says on that page that
You must be aged under 45 when we invite you to apply for the visa.
You can still apply for the visa if you turn 45 after we invite you to
If you turn 45 after you submit your EOI but before we invite you to
apply, you will not be invited to apply.
Contrast this with the 186 visa age requirements which say:
Usually, you must be under 45 years of age when you apply.
You are exempt from this criteria if any of the following applies to
- you are nominated as an academic (university lecturer or faculty head at Academic Level B, C D or E) by a university in Australia
- you are nominated as a scientist, researcher or technical (scientific) specialist at ANZSCO skill level 1 or 2 by an Australian
scientific government agency
- you currently hold a Special Category (subclass 444) or New Zealand citizen's family member (subclass 461) visa and have worked for the
employer who nominated you, in the nominated position, for at least 2
years in the last 3 years immediately before you apply. The 2 years
excludes periods of unpaid leave.
Note that Australian visa eligibility criteria are often different for citizens of New Zealand. The page for the 189 visa for example makes no mention of age. (Though that may come into play in the sub-requirements listed. Do your own research, obviously.)
Have a look at the visa list page on the immigration website.