In fact, developed nations in Europe and the Americas are importing vast numbers of low wage workers to do low value per hour jobs where humans are still better than machines. It continues in the face of a prolonged severe recession.
So the emerging problem of no available work to do does not seem to be materializing. In fact, it's hard to imagine. Does every public park and private garden in your city look as clean, inviting, and pretty as it could possibly be? Is every street paved smooth without cracks and potholes? Does a bumper crop of fruit never go bad for lack of harvesters? Is every home clean and freshly scrubbed all the time and is a freshly cooked meal always waiting? Does every corner have a crossing guard during school hours?
There's a lot of intellectually undemanding work to be done in a utopia if we want to pay for it.
Right now, we import foreign workers to do cheap labor but we could take unemployed citizens and subsidize their wages to have our own do those jobs instead. The unemployed citizens are living on private or public welfare and social insurance today but they could be working. We could kick in public money to make it possible to pay a legal wage with benefits for the jobs undocumented workers do and the employers would prefer to stay legal by hiring citizens. We could even document the undocumented and add enough subsidies to employ them too.
Economic change driving social change is disruptive. But let's remember that technological productivity is a positive change that makes us wealthier.
Unemployment that follows from energy shortages or global climate change, if those ever get serious, would be a much bigger worry.