We literally collect people and bring them across the Mediterranean and the Channel. We then never make them leave when we discover they aren’t genuine asylum seekers. In all honesty, Putin would be wasting time and money, as we are already our own worst enemies in this front. He couldn’t orchestrate nearly such a successful system of mass migration.
Besides, on the U.K. we get about 1,000 illegal dinghy arrivals a week, about a similar number by other routes (lorry / plane) and 21,000 legal migrants, only about 3,000 of which will pay taxes here. We set our own immigration policies, so we have to take responsibility. Illegal migration really is the tip of the iceberg.