The NMW for a start should apply to all workers & should be significantly higher - it's nowhere near a living wage.
The point about social mobility is that the government should support people who are trying to move to different parts of the UK to find work, currently it doesn't.
But even taking that into account we still have a jobs deficit.
I had to move myself for work, but not everybody has the opportunity, the job offer, the income or the savings/support to move (it wasn't cheap).
I can understand wanting to get people into productive work - really.. I really can - but then let's have a frank & honest evidence based discussion on the barriers to work & reasons behind long term unemployment (outside of simply calling them lazy & suggesting punitive measures which will most likely cost more in the long-term due to reduced social cohesion or increased crime).
Currently we get right wing ideological sound-bites to support changes to our welfare state - I'm sorry but that's not good enough - I want to see some evidence to back these assertions up, more so when they are likely to cause the opposite of the desired intention (if you take into account various studies on human behaviour).
Aside from the arguments about NMW increasing inflation, NMW does apply to all workers over the age of 18, the point being those under 18 are supposed to have less financial liabilities, although I don't necessarily agree with that at all!
You mention it not being cheap to move but do you have family, mortgage, car etc? Thousands/hundreds of thousands of ex students move each year to other areas of the country, a significant portion of those without jobs before they move. The point being a single long term unemployed person is unlikely to have a mortgage, car and other financial constraints, much like a student after university. It's a different situation. If hundreds of thousands of students can do it why not unemployed people? Especially if we can persuade them with a bit of effort they can have a better life, which is what a lot of the charities dealing with long term unemployed are trying to do.
I'm certainly not suggesting punitive measures, what I have been suggesting all along is this cap designed as a cap for those using disproportionate resources and as stated the cap doesn't affect many people. I don't think many begrudge giving people a couple of hundred pound a month and a free room while they are unemployed, however many people begrudge the idea of the long term unemployed living in potentially nice flats/houses in central London, all paid for by people who have to commute an hour plus a day, potentially living in a shared house or with parents, to work, right beside where these people are living, because they can't afford a even a room in that area.
Its all very well stating sound bites from the "left" but you still have to answer the questions.
From time to time all companies need to look at their finances, decide they need to make cuts and cut to reduce overheads, government is no different and the cuts do seem small when taken individually, however when added together with cuts in military, research funding and other locations you get big savings, this specific cap hardly affects anyone, especially compared to some of the other cuts that have affected more (redundancies and base closures due to military cuts, let alone the knock on to British business that supplied those soldiers and bases).
Edit: that's aside from the stupid price of housing, which is a whole bother discussion. Just slow the bubble to bursts and reduce prices. It's all very well stating we should build more houses but we all know they will be built in the south, I'm sorry but I'm going to go all NIMBY but the south east is one of the top 3 densely populated areas in the world already and I've seem thousands of houses built round my childhood town in the last 20 years, places I used to walk in the countryside now a sprawl of ugly cramped houses, that's aside from the other 20000 houses they want to build on the other side of town, which currently has a few small villages and farmhouses... Anyway... Rant over...