Interesting Cross Domain Canonical Quirk...
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We recently ran cross domain canonicals for 2 of our websites. What's interesting is that when I do a search for ""site:domain1.com "product name"" the Title in the SERPs uses the Domain Name from the site the page has been canonicaled to.
So the title for Domain1 (for the search term above) looks like this: Product Name | Keywords | Domain 2
Interesting quirk. Ha anyone else seen this?
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That's the way it should work. When you set up a cross domain canonical from a URL on domain 1 to a URL on domain 2, you are telling the search engine that you want the content on site 2 to be indexed rather than the same content on site 1. The page content on domain 1 is probably not in the index for search results anymore, but the canonical tag ties the content on the two domains together.
In your example, does the search results link to the content on domain 2? That's what I would expect.
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Interesting quirk. Ha anyone else seen this?
Working as intended

As Laura said, when you canonical (a) to (b), you expect (b) to become the dominant page / site.
-Andy
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What was the result you were expecting?
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The SERP did link to the correct (canonical target) domain. If the canonical tag is on domain1.com/product-a, the SERP was correctly pointed at domain2.com/product-a.
Because the page on Domain 1 is supposed to be de-indexed, I was expecting not to see the page at all. This is my first crack at cross domain canonicals. It's an interesting way for Google to handle it.
BTW, from a rankings perspective, the cross domain canonicals were extremely productive. Domain #2 got some huge rankings increases.
I've been tracking the results closely. I should publish the results when I get a chance. The most important result is that the keywords (+/-700) associated with the canonicals  improved by an average of 22 positions over the higher position prior to the canonicals being implemented.
What I mean by that is for a keyword (ex: "widgets"), Domain 1 was Ranked 46, and Domain 2 was ranked 57, our average improvement was to position 24, which is 22 positions better than the higher ranked domain (in this case, Domain 1).
Rankings improvements for keywords already on page 1 or Page 2 increased by an average of 2.5 positions over the better ranked domain.
What was really cool was that when we canonicaled in the "wrong" direction, where the keyword ranked higher on the domain that was getting the canonical tag, the results were indistinguishable from the results where we canonicaled in the "correct" direction.
So, in this case, if a keyword ranked higher on domain1.com, and we canonicaled to domain2.com, the average ranking increases (from the higher ranking position) were almost identical to using canonicals in the "correct" direction (from the lower ranking position).
These are both ecommerce sites with DAs of +/-40.
What was also interesting is that Google accepted the canonicals in cases where our product descriptions were markedly different.