If a GFCI starts tripping with an appliance plugged into it and you replace the GFCI and the tripping continues, you should assume the appliance is leaking too much current to ground from either the hot or neutral wires.
To verify this, if you care, you need to be able to read the difference in current between the hot and neutral wires passed the GFCI and feeding the appliance. If you separate the hot and neutral wires from the grounding wire in an extension cord and pass the hot and neutral only through a clamp-on ammeter that can display 4 milliAmps then you can look for excessive leakage.
The problem is finding such a beast.
Here is what I have done:
1) I made three separate coils of 10 turns each, black, white, and green, and taped them to hold their shape.
2) I connected one end of each coil to a plug and the other end to a receptacle, matching the colors.
3) the coils are side by side.
I can use a clamp on meter around the black and white coils to see the difference in the currents since when the current is flowing one way in the black coil it is flowing the opposite in the white coil. This difference is what the GFCI is looking for. I have multiplied the currents by 10 as far as the clamp-on knows and so have a chance of seeing milliamps with a meter I can afford.
Picture to follow in future post.