Sometimes it doesn't, which can mean that people really need to move out and store their furniture in the removal can while they sleep elsewhere overnight. And if the chain is a really long one, they try to split it, which can mean if the chain is longer than ten, say, the people in the middle will be persuaded to move into rented accommodation for a bit if possible, as we have had to do, just to get the ball rolling.
It's a truly awful process, but sometimes it all goes smoothly, so good luck.
It starts with the people at the very bottom of the chain, maybe first time buyers, having their mortgage in place and their paperwork also. Then their solicitor will contact the next one up the chain. They will have arranged a provisional date in advance. Once you have all agreed on that date, it's best to check that your own solicitor won't be away on holiday, they seem to be away an awful lot. The solicitors and estate agents will all be checking up on each other to ensure that all paperwork is ready and signed and no worms about to crawl out of the woodwork, but even so some things slip through, and there are always buyers hoping to knock peope down a final £5000 - in our case because we owned the pavement outside at our last house and a lorry reversing dented a bollard... The cost of replacing it would have been £4000! I didn't mind a dented bollard, but they did (strange that it's still there 6 years later).
So, depending on the length of your chain, the phone calls start early in the morning, so be contactable. Don't have your phone cut off before you move - and sometimes the phone companies switch off a day early! Make sure everyone has your number in case things go wrong. Go ahead with packing your stuff. We lost our house the day we moved out, so DBH was sitting on a stoll in the sunshine phoning B&Bs and prospective house lets as our packers carried on, we had to move because he was about to start a new job, and if we had unpacked - we still would have had to pay the removal company! So we packed and took the keys down to our estate agents, for the next people to collect.
Every move is different, but we are now in our 12th house in 20 years, so ask away!
Stay on top of your solicitor, they are overworked, so make yourself a pest, but nicely, so they sort your stuff out first.
Estate agents quite like it if you don't contact others in the chain in case you do deals behind their backs, but we found it invaluable to stay in contact so that we had advance warning of potential divorces/job losses/change of heart.