A history of a mailbox(es) and their home database is needed. This is typically needed for backup restoration purposes.
Move Request data is typically not saved. If the move is not performed via the ECP or if the migration batch data has been removed, the logs are no longer easily accessible.
Open the Exchange Management Shell and run the following command - replacing $emailAddress with the user's email/alias.
(Get-MailboxStatistics $emailAddress -IncludeMoveHistory).MoveHistory
It is worth noting that this will return all of the data that was originally available during the move request, most of which is not what you are looking for. Pipe the above to Select-Object or Format-Table to get the data that you need.
(Get-MailboxStatistics $emailAddress -IncludeMoveHistory).MoveHistory | Format-Table StartTimestamp,CompletionTimestamp,SourceDatabase,TargetDatabase -AutoSize