I'm surprised freedom of movement is top of your list? Why do you think it will be easier after Brexit? I would have thought it would be harder
Again it's quite ironic in a certain way. You should check EU history. In the whole, the UK (well the government- normal people have relatively little to do with it) until a while ago, was much in favour (unlike e.g: France, Belgium, some of the Mediterranean states...) of pushing the further expansion of freedom of movement for workers/even the further expansion (wide) of the EU.
An example: the European Council of Berlin in 1999 (concerning a date of accession); two visions: 1) Immediately set a date, regardless of the progress that the countries had made (technical, financial, etc...): mainly championed by the U.K and Sweden. 2) Make the date dependent on achieving individual (per nation) actual benchmarks of progress: mainly France and Belgium. To make a long story short U.K won: in 2001, even though the E.U commission was reluctant, and there was also resistance growing in Germany (and was still the same in France, Belgium and a couple of others): it ended up being June 2004. And now a lot of them are blaming the EU for expanding too quickly and taking on countries with too different living circumstances...
Or if the Brits would have gotten their way Turkey would have probably been in the E.U right now. They championed them joining for ages. Now it's a threat though: "Turkey might join the EU !" - no they won't; too many outspoken opponents, granted it's not entirely fair since they have had an association agreement since 1964; they should probably do the honourable thing and end that agreement. Only a short while ago, I vividly remember Cameron insulting the French and German prime ministers because of their resistance to let Turkey join the Union ('double standards').
The above is all forgotten: it's everybody else's fault. They did democratically elect their leaders though.
Didn't always think it through though, e.g: let's look at the EU-8 accession (probably important for the current situation).
The U.K was one of the few countries that immediately opened up it's labour market for workers of the EU-8 (there was only a minor administrative obligation). They expected 14.000 new workers; they got a whole lot more than they expected. The effects were positive - did increase the pressure on housing (so the response would be to rectify that) etc ...
Difference in approach: most countries in and around the Benelux implemented transition-provisions. In Belgium and the Netherlands, initially to protect the labour market because they feared that the domestic workers would be overrun (motive changed quite quickly - wasn't credible any more; it was quite clear early on that the workers took complementary jobs that domestic workers weren't keen on-positive effect). So basically controlling the flow of workers/level of immigrants.
The U.K implemented "transition" provisions because they feared they (EU8 workers) would plunder their social security (benefit tourists if you will; that work very hard and take jobs that aren't popular). So they weren't bothered about the amount of immigrants. They just wanted to be able to exclude them from all sorts of social security. Couple that with the the very negative portrayal of EU8 migrants in the U.K (lazy, wanting to take advantage of social security, when they commit a crime it is in everybody's importance to expressly stress the nationality of the person involved). Those provisions were not very friendly; when the 8 countries signed their accession agreements (and the entire acquis and what all), there was no mention of that.
So they profited from a new flow of foreign workers, without actually giving them the according rights that come with being a worker. The entire thing was a travesty. The impact of those workers was quickly, objectively seen, as positive. Still they used those discriminating provisions until the last possible moment (I think until somewhere in 2013). The last extension required that they were able to proof a serious disruption of the British labour market. There was non, the EU commission still granted it (brilliant work from all involved). Tbf during that entire period the performance of the commission was toothless throughout, they basically asked politely: can you please stop discriminating and respect EU law: U.K said no (rinse-repeat). The U.K courts were equally helpful, "yes we're discriminating but ...". Brave people who went to the U.K in those days; effectively second-class citizens compared to those of the EU-15 and domestic workers.