Making spice pastes with a food processor I've had call to make spice pastes by using a food processor. Rather than using the full-sized bowl, which is way too big, my food processor has a milling attachment designed for coffee beans. However, whenever I try making spice pastes in this, it only gets so smooth before becoming clumpy and the blades effectively spinning at nothing. (Ginger and coconut are particularly bad at not reducing down further than grit.)

Adding more water usually helps to make a finer paste, as it forces everything together somewhat and causes the mixture to act more as a liquid than a powder. However, that makes the paste too wet (it's a paste, not a sauce, after all!)

Is a food processor/mill not the right tool for the job, or am I just doing it wrong?
It sounds like you are doing it wrong! :) Grind spices in a spice grinder first (a rotary coffee grinder works fine for that). See How do you finely grind coriander seeds? 

Garlic, ginger, fresh herbs and coconut are fine in a small bowl of a food processor, but they need some time and a few restarts after cleaning the sides of the bowl with a spatula. After processing the spices and other dry things in the rotary grinder, you can add the spices to the moist things after they have been processed. The food processor will be fine to mix the dry and moist parts, after they have been ground/processed separately.