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The electrons that leave the tip of the needle don't get very far
on their own, certainly not as far as a millimeter. They interact
with air molecules and do not retain their identities.
In a vacuum
things would be quite different. "Field emission" may not be the
best term to characterize phenomena in air at centimeter distances
from the source.
I would expect overall neutrality of the material in the stream. It
is composed of ions of both signs. The electrons (which predominate
the negative ions) are the more mobile, of course. The same thing
happens more dramatically in a lightning bolt. You have a large
current limiting element in your circuit; that's the difference.