The velocity of money is a function of taxation. The less tax, the greater the velocity. At 100% tax, the velocity is near zero or if someone spends a dollar and the govt. takes it all, velocity goes near zero. At 50% tax, a dollar is spent, the govt. takes 50 cents and the other 50 cents goes on to be spent again.
Since illegals pay about the same amount of tax as a citizen at the same income level, I would expect the velocity of money for illegals just as fast as for citizens in the same economic strata.
Velocity of money refers to the number of times that money circulates within the economy thus having a multiplier effect on the money supply and a corresponding expansionary effect on the economy.
Money that is sent out of the country does not circulate in the economy. The immigrants get it and then it is gone. It is off some where in a foreign land circulating and generating wealth for them rather than us.
That differs sharply from the same wages which are paid to a lower income American which are spent in the local economy and then the recipients spend it again on things like wages and cost of goods and rents and then the recipients spend it again. One dollar can circulate around in the economy numerous times generating profits and wealth for all who touch it.