For a blog newly published to a different URL - either a non BlogSpot URL, using custom domain publishing, or simply a different BlogSpot URL - the first few days under the new URL are critical to the success of the blog. Some blog owners find, to their horror, that their blog suddenly has no page rank, and has dropped out of visibility in the search engine results.
I bought a domain from Google, and my blog is now invisible. Obviously, this is another swindle by Google, stealing my money!These blog owners have not done any research, about the custom domain migration process, or about any issues related to renaming the blog. The successful migration process starts with getting the blog, under its new URL, indexed by the search engines.
One of the most useful tools, that you as a blog owner can use, is a sitemap, defined using Google Webmaster Tools.
The default sitemap, for a new blog, indexes the most recent 26 posts. For most blogs, you will seldom publish more than that between each indexing pass by the search engines - and a 26 posts submission is generally sufficient.
When you re publish your blog under a new URL, you want all posts reindexed immediately. To get all posts reindexed, you are going to need a more robust sitemap complement. The new sitemap will require a new entry, in Webmaster Tools.
You want your blog, under the new URL, to be visible under that new URL as soon as possible, so setup a proper sitemap - then get to work publishing new posts, and increase search engine activity. And, use Jump Break for all new posts, to focus search engine attention on the individual posts.