It sounds like PME; if it's a 30mA RCD, I wonder if there's enough general leakage (washing m/c, immersion heater etc) to be near the tripping current, and the lawn mower pushs it over the edge?
If it's a 100mA RCD, then there would surely have to be rather a lot of background leakage to be near it?
Phil, the pictures would help, if you can lay your hands on a digital camera. In the meantime:
Does your RCD protect all circuits, or just the sockets? Can you see what letters and numbers are on it?
Have you got, for example, outside sockets/water features/pool pump/fountain/supply to shed/anything that might get damp? Or a pipe to a garden tap? The reason I mention the washing m/c and immersion heater is that in my limited experience, slight background leakage comes from the "watery" appliances, and these small leakages can add up and take you near the tripping point.
BTW one of my relations had an electric mower with an old flex, it looked all right, but when it ran across wet grass, it tripped the RCD. I didn't see it but suspect old rubber insulation, which gets little cracks if left out in the sun. Doesn't explain why the trip is jumping from house to house, if there's no earthing spike, though.