Does StartMessageMoveTask support SNS DLQs and cross-account targets?

Here are the answers for your queries. 1/ The DLQ redrive works if the DLQ has received messages from any other SQS queue. If the DLQ has received messages directly from Lambda or SNS, the redrive to the source or custom queue is currently not supported. As a workaround, you can create a new queue and use the existing DLQ as the dead-letter queue for that new queue. After doing so, you can re-drive messages from your DLQ to any custom queue in SQS. 2/ DLQ doesn't support cross-account. DLQ must be under the same region and same account, so does redriving messages back to the source or custom queue.