How to process user offline message and send to certain party and consider the availability

Some requirement: User will send message to our system and we will collect and store somewhere. In the night, or maybe once in a hour, we will go through all these offline messages and send to the destination the users specified in the message. How to do these gracefully if considering below: Web server load balancing, continue to proceed if one server failed in processing and the other continues to proceed? Avoid the processed messages when another server continue. Any recommended way to do that? What nice framework or practice?

Thanks!