I don't know if there is a standard [behavior] for this case.
I don't think there is. I have seen both screenshot galleries and full-page duplication, as well as simply links to the original pages.
The question is, should I allow those pages to be indexed by search engines?
If you use screenshot galleries or link to the original domain, this shouldn't be an issue.
If you do host a duplicate of the content, did you get permission from the client? Depending your agreement with them, you may risk violating a given countries copyright law. Personally, I would suggest introducing in a clause for all your contracts which clearly indicates what content will be used in your Portfolio and how. Google specifically may remove duplicate content if it runs afoul of US DMCA provisions or Google determines it might be for ranking manipulation.
That said, it is a pretty hard scrabble sometimes in design, so the choice is yours. I would at the very least be putting links back to your own domain in each applicable page footer (as a "design by" credit) unless you agree with a client to do otherwise.
Should I use [noindex] or not?
Each search engine may handle the tag differently, but I would not unless you really don't want that page indexed. It basically indicates a "no go, humans only" for respectful bots. You would likely only want to use this tag if you want to ensure that the duplicate page NEVER appears in Google and is unlikely to appear in other search engines.
Regarding the whole concept of archiving web page "snapshots", this is the entire premise of the Internet Archive Wayback Machine.