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Hi Eric, As I say I'm familiar with the general principles of redirects, having applied them before with other clients. Unfortunately it's the industry-specific knowledge I need. Link resolvers are regularly used in academic publishing to point from, for example, library holdings pages to an ebook or a publisher's platform, at which point temporary redirects make sense for various reasons. My concern is that in using them to point from our old site to our new site will cause us problems.

Ben, I'm assuming that you have 301 redirects properly set up for all of those page. Then, make sure  you've used the Google Change of Address Tool to properly tell Google that you've moved from one domain to another. If you are consolidating domains, then you need to verify all of those sites in Google Search Console and then use the Change of Address Tool. You don't mention how long it's been since you've set up those 301 redirects, but it literally can take months before everything is straightened out in Google. One thing you can also do is to look at the site's log files to see if Google is crawling those pages--it could be that they're not crawling. If they have crawled, then it might just take time before they're indexed.