As of 3 December, 2015, Tweepi's 'Not following you back (formally Flush)' has been removed at Twitter's request. I want to check how much is there which we can not integrate or Twitter API does not provide it. Please do not award until I complete my analysis, I do not mind to give you a free consultancy before start the work.
Due to the latest Twitter updates to their 'Automation Rules and Best Practices', we are forced to disable multiple selections and 'Quick Actions'. As a result, you'll notice that the checkboxes have been removed and you can now just select one user at a time. The new 'Default Button' emphasizes that button on each user row, making it easier to follow/unfollow users. - So this is another one from tweepi which we will have to note.
Now, you are right that digging the application is required and need to check what API is required to complete the action. For an example, I have tested that all the list is not updating in real time, have you checked it? Just follow/unfollow an user and try to refresh the dashboard or list of followers, you will not get the new one. So, I am assuming they are using some backend service which is synchronizing with Twitter. I am looking into it and will get back to you with a solution.
To be honest, I used this application as a Free user so if you have a premium account then please share me so that I can dig in more details.
I will wait for your more update or would be better if you can come for a discussion.