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Ya, and this will create more incentive for other people to become plumbers and plumbers to immigrate from elsewhere in order to profit from the labour shortage in the industry. The same holds true for neurosurgeons, veterinarians and every other job which requires a particular skill.
And round and round it goes, more dodging! So what replaces the lost labour force?
Yes, the force lost to the richer country.
Since Tom's answer did not satisfy you, you must be referring to the total loss of population, which would mean not as many plumbers would be needed.
What the loosing country needs to concern itself with is brain drain. A country can easily replace the plumbers lost from it's own population. Neurosurgeons may take a bit longer. But this takes the mass out of your "mass immigration" issue.