Having cleaned and rebuilt a mass produced movement that appeared to be in good condition, I find that although it will run for 14 days, it suddenly begins to lose time after 3 or 4 days running. When it starts to lose time, it loses time rapidly. The movement dates approximately from the 1920s but is not marked with any maker's name. Does anybody have any suggestion of what might cause this?

You don't make mention if this is a spring wound clock or weight driven. I bet it has a spring....I would check to see if you can find a replacement set of springs. Measure teh width and thickness of each spring and see if your supply house has one to match.
Likely it has lost its "spring" or power. Since it becomes weak as soon as it starts to unwind and the rate drops.If you would rewind it at 3 days, maybe you would experience no loss.
Moses in Waxhaw NC

Thank you for your response, Moses. The clock is indeed spring wound. I am new to horology but I wanted to check the springs when I rebuilt this movement so I dismantled the barrels then cleaned and greased the springs. I also had to build a Joe Collins style mainspring winder to get them back in! I wonder if it is possible that through inexperience I have over greased the springs and caused an uneven power release in the barrels? Otherwise, I imagine you are right to suggest that the springs have lost power and need renewing to achieve a full 8 day action. Curiously enough, having been running for a couple of months, this week, the clock did not start to lose time until the sixth day - so maybe it is beginning to free itself up?

Well, the fact that is better after new grease shows that it very likely is the springs. That way it is, just wind it every 5- days and you have solved the problem! But in the event you would like it to run the as it should a new set of springs is likely to make the difference.
Hope this helps you decide.
Moses -in the jungles of Waxhaw