Here is a paragraph explaining Procedural Memory (from the TOEFL exam)
The memories people form when they learn names and facts are different from the memories they form when they learn how to perform a task.These memories of performing particular actions are called procedural memories. Procedural memories are memories of the process of performing a task that become automatic with practice. Once a task has been practiced, or repeated many times, procedural memories are established. These procedural memories allow people to perform the action automatically and to recall it relatively easily many years later.
So the focal word is memories; memories become automatic with practice.
But can I write:
Procedural memories are memories of the process of performing a task that becomes automatic with practice.
I want to write this because "a task that becomes" looks better to me, also because maybe a task can become automatic?