indicating something proceeds in some process advancing its state. It is therefore often described as having a 'directionality' when compared to 〜ている.
氷が溶けつつある The ice is melting. (the process of melting is ongoing)
日本では物価が上がりつつある。 In japan, prices are rising.
It really needs to be a process with can 'proceed', and not just any verb.
リンゴを食べている リンゴを~~食べつつある~~