Are you using a keyword to directly trigger an automated message campaign (AMC)?
If so, perhaps you noticed that it can take up to 3 minutes to process the first message.
If this first message is a critical, text auto-reply, this may not be desirable. Given the way AMC currently works, this is unavoidable when using the above hypothetical setup.
To view which triggers you are using, in the Campaigns page, click the "pencil" icon (edit) next to the Automated Message Campaign you want to see:
Then click "edit triggers" to see or edit the triggers you are using for this campaign.