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It looks like you've got a backups folder in your site's root with very large backup files. These cause a timeout while they are being transferred. After completing the first two steps of the Site Transfer Wizard, go back to Akeeba Backup's Control Panel page and make sure the selected profile is "Site Transfer Wizard (do not rename)". Click on the Files and Directories Exclusion and exclude that directory. Go back to STW, click Next on the first two steps and begin the transfer. The transfer should now work.

There is no size limit per se. It all depends on the PHP timeout limit, Apache's timeout limit and the speed of FTP uploads. Especially the latter can have a very big variance, so it's impossible to say that files of such and such size are too big to transfer.

If you want to easily transfer your site, I recommend another approach. You can use the Upload to Remote FTP Server data processing engine and a part size of 2-5Mb for your backup archive. For more ideas about transferring your site, take a look at my presentation for Joomla! Days Netherlands 2012. The last few slides give you a few different ways to do it.

There is also another way, not described in the presentation. If you are using Amazon S3, you can backup your site to Amazon S3. Then, using Akeeba Kickstart Professional 3.5.1 on the target server you can download the backup archive from S3 and proceed with its restoration.