well you've delved into potentially confusing territory.
a cell is constantly synthesizing new proteins and is never, ever at rest or equillibrium, we call that "death".
However, you specifically asked about muscle synthesis.
One feature of homeostasis is a constant osmolarity, which is dependant upon the amino acid content of the blood. This means that amino acid content of the blood should remain relatively constant.
if this is the case then when you have no food in your alimentary canal, amino acids are degraded from body protein and released into the blood stream, this is catabolism. Eating prevents this process.
Now specifics about muscle synthesis. I'll use an example of DOMS. With DOMS you can have soreness for days at a time, with the support and evidence that DOMS is more a cause of local edema and swelling caused by muscle tissue repair, it is evident that muscle growth does not simply begin and end near the workout period.
The literature even states that within a 48 hour period of the workout your body responds to dietary protein in different ways than normal, read: 'youre body is in a state of synthesis'.
The healing of muscle tissue after a workout, is obviously dependant upon the strain placed on the tissue, but ignoring that. No, the increase in muscle synthesis after working out does not halt within a couple hours.
Keep in mind i say increase because there is always a backround "noise" of small amounts of synthesis as well as very small amounts of catabolism- both dependant upon a variety of factors more than just dietary protein intake.