Although I have been a reviewer for 8 years now, I never received an invitation from Journal yyyy to review a manuscript for which I’ve already been a reviewer for Journal xxxx.
I am curious… How do you approach the review of a manuscript submitted to Journal yyyy with no evidence of (even minor) update from your previous review for Journal xxxx (obviously, this manuscript has been rejected for publication in Journal xxxx) ? Will you ask to authors the reason(s) why they decided not to change their initial version of the manuscript or will you copy-paste the same general and specific comments ?
Le Physiologiste 
Nothing wrong with declaring previous review for another journal. Letting the editor (and authors) know this is good all around. Quick decision for editor, authors will learn from it. If they’ve done nothing to it, that’s a sign of laziness if you recommended lots of changes. If they made those changes and you think it was unluck to have been binned first time around, both parties will appreciate you saying so. OK to re-review, and best to say that’s what you’re doing.
This seriously pisses my shit off and had it happened to me this year. I sent the exact same comments to the authors and let the editor know that I had previously reviewed the paper and had given them the attached comments previously.
I think that I would do the same…
I agree- nothing wrong with reviewing it again but just be up front about it to the reviewers of the second journal. Did the authors make changes to the paper? It is pretty normal that it would get submitted elsewhere if it was rejected from the journal that you had first reviewed for. Perhaps the second journal has a lower impact factor/different philosophy….?
I agree that it is normal to submit a paper elsewhere if it is rejected by a journal in the first place and that there is nothing wrong for a referee to reviewing it again.
However, if the authors of the rejected paper submit it to another journal without making any changes and this paper is reviewed by a referee of the original submission, I think that the authors are in trouble.
In such a situation, I think that it would be clear to the authors that I was one of the referees of their previous submission…
I said that given the absence of changes in the manuscript, I did not need to update my review. I then pasted my previous review.
Our lab has done this before. It was frustrating to see they didn’t fix any glaring issues with the manuscript before they shopped it to the other journal that we review for.
I am sure that it is frustrating especially if we do a great job as a reviewer and put a lot of time on that review!