Why does my Avast antivirus keep on downloading something even after the entire setup process is complete?
Without more details it is hard to guess.
What avast version and build number are you using ?
What is it downloading, from where and how are you monitoring this ?
As a resident antivirus it is going to be scanning your traffic between the internet and your system (browsing, emails, etc.), but that is traffic you initiated.
Virus definitions (streaming update) happen every few minutes.
Hopefully you can see why we need more information.