I think the finite size is a red herring; with ideal sources/sinks and particles it still doesn't "work".
Incorrect approach: 100% of rays from from blue focus go to the red one; 80% of rays from red focus go to the blue one; 100% > 80%; therefore imbalance. Incorrect because we must look at absolute numbers, not percentages, when saying the flow from blue to red must equal the flow from red to blue.
Incorrect approach: Initally both foci emit 100 particles, and during a time period epsilon, 100 particles go from blue to red, but only 80 go from red to blue. True so far, but Incorrect approach because the paths are not all the same length so the travel times are not all the same. When all paths are equally full of particles, the numbers arriving at each focus will be the same.
Correct approach: Short path takes T1 time units to travel; long path takes T2. At equilibrium, during a time period epsilon, 100 particles will leave blue, of which the fraction that hit the tighter-curved mirror will arrive at red after T1, and the fraction that hit the longer-curved mirror will arrive at red after T2. During the same time, 100 particles will leave red; a fraction of those will go to blue in time T1, and a fraction will go back to red in time T2. I'm too lazy to work it all out right now, but that's the correct approach.