Pick up/take the ball and run with it
Definition 1 of 2
Pro Tip 1/3Proactive Attitude
The phrase often implies that someone is proactively taking on responsibilities without being asked. 
As soon as the lead designer quit, Mia picked up the ball and ran with the new project.
Pro Tip 2/3Positive Progress
It suggests that not only was a task continued, but also considerable progress was made. 
After our coach fell ill, the captain picked up the ball and ran with it, leading the team to victory.
Pro Tip 3/3Self-initiative
Indicates an individual's initiative to take control of a situation without relying on others. 
Lena picked up the ball and ran with it, organizing the event when everyone else was hesitant.