Fred,
I see two possible situation:
1) The problem GFCI is defective and needs replacement.
2) That GFCI is protecting other downstream circuits and is detecting some minor GF problems from downstream. The espresso machine may then become the "straw that breaks the camel's back." GFs are accumulative and the GFCI is effectively summing all of the GFs on circuits that it controls. Note that the GFs may be real problems or just so-called "nuisance" GFs.
Somebody really needs to market a plug-in device which measures the level of GF for any device plugged into it. If anybody is aware of a reasonably priced device that does this, please let me know before I start building one.
Such a device should be pretty easy to put together with a portable GFCI, a 120V outlet splitter which makes it possible to plug 2 loads into the portable GFCI, a potentiometer for inducing a variable and measurable GF, a fixed value resistor to limit the induced GF level, a multimeter for a one-time calibrating of the potentiometer, etc.
Fred, you probably should buy or borrow a GFCI tester which has user selectable GF settings. MCM Electronics sells such devices and I'm certain that there are many others who sell this item. This makes it possible to test your problem GFCI and to determine what level of GF plugged into that GFCI will trip it. You can perform the same test for the other GFCI in your kitchen.
Also, remember that GFCI units are relatively inexpensive and very easy to replace. I paid $4.49 each for the most recent batch that I purchased. To simply replace the GFCI and see what happens may be the most pragmatic approach.
Let us know what you discover. We all appreciate the feedback.
Good luck, Gideon