How search engines react when you move hosting?

Selected answers from the Dumb SEO Questions Facebook & G+ community.

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  Steve Gerencser: In theory, the old host will clear that spot on their server so that they can use the space for paying customers.
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  David Kutcher: Yes, they could find and index it, especially if it was on a static ip address.
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  Ammon Johns: Google became a domain registrar themselves many years ago, and do their own internal DNS lookups, just to make themselves immune to losing time if other DNS services are down. This tends to lead to the obvious conclusion that Google index by both domain name and by IP address anyway, with their own internal DNS lookups. It shouldn`t create any major issue for you, but yes, be certain that Google will know of both versions. For iron-clad safety consider not only cancelling the old hosting, but also implementing a sitewide redirect or 410 in the interim while the final time expires.
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  Jody Nesbitt: Neil, honest answer, yes it can be accessed, assume the worst case, you had a few absolute links in your old site referencing the staging path or temp path from before you migrated, now because this is an absolute link, its accessible independent of the NS. I have seen this countless times where we migrate sites for clients and everything works fine until they turn old account off, then suddenly broken pages and images appear, best is to backup old, turn it off (delete the top level public folder) and test your site, if all is OK, then safe to remove, and remove sooner rather than later cause these things have a habit of getting forgotten and auto billed before expiry
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  Tony McCreath: I have a client that switched hosting. I still get emails from the old copy of the website, and have no known way to access it. So host do leave old accounts hanging around, and sometimes with alternate ways to access them, like special sub domains or ip addresses.