Hi njjl,
I will try to explain the issue. All-in-One WP Migration copies over the plugins, themes and media files. When the files have been copied over, it transfers the database.
When the database is transferred, all of the plugins and themes are activated. During this activation step, you receive the message “Fatal error: Allowed memory size of 33554432 bytes exhausted (tried to allocate 953384 bytes)” and the plugin hangs at the restore.
The database is actually restored and the website content with plugins, themes, and media files are transferred but your server or its settings are not configured to handle the website you just moved.
There are a couple options: you can upgrade your server to one that is more powerful if that is the issue, or you can update your web server configuration to allow for more memory. WordPress recommends at least 256MB of available memory and looks like the server failed to proceed after reaching 34MB (that’s what 33554432 bytes equal to).