I doubt there is a thing called as pending messages in Kinesis. Now we have a shard in Kinesis where all the incoming stream gets stored.
Your consumer application has to always be in running state and keep track of changes in your stream. The application will continue to poll "Shard Iterator" in the background, thus you will be notified about the new data when it comes.
Think of Kinesis as a FIFO queue and the messages will disappear in a short time if you don't pop them.
I hope this helps.