Before going further, I think it is worth pointing out that trapped ions have any-to-any rather than all-to-all coupling. With maybe a few exceptions, no one is doing anything more than two qubit gates.
The basic idea of trapped ion quantum computing is generally as follows. Typically, you encode the qubit state in the ion's spin (hyperfine/Zeeman qubits). To perform two qubit gates, you couple the ion's spin to it's motion by using focused lasers. Without those lasers, there is virtually zero coupling between spin and motion, which is the big feature of ion trap quantum computing. In fact, this lack of coupling is what makes two qubit gates rather challenging technically - you have to use a considerable amount of laser power for a relatively long time to do gates. So the way crosstalk errors occur is actually quite local - the nonzero spot size of the lasers means that neighboring qubits will undergo unintended gate operations as well. However, since the laser intensity is gaussian in space, the effects on even next-nearest neighbors are exponentially small.
However, it is worth pointing out that gates can and do introduce motional heating if they are imperfect. The idea being that a perfect gate returns the motional state of each ion back to where it started, but imperfect gates leave a bit of motion left over. You may think that this motional heating will mess up gates coming afterwards. However, a main selling point of the Molmer-Sorenson gate with ions is that it works regardless of the initial state of the motional degrees of freedom. So while this heating does affect gate fidelity, it does not cascade and propagate errors in the way you might be wondering.