I've been following a discussion in chat, and I've noticed that the replies to my messages haven't been showing up in my inbox. One of the replies that I got (which was 12 hours ago) didn't show up in my inbox for some reason. Is there any way to change the chat settings so that all replies will appear in my StackExchange inbox?
If you're actually there in the room and dismissing the replies as you continue chatting, then they won't show up in your inbox. Chat messages are only pushed to your inbox if the reply has been waiting for you to view for some time (I don't know the exact amount) or you're completely absent from the room. If you've already viewed and dismissed it, there's no reason to notify you of the reply and there's no way to change this functionality.