As last seen, the Dursleys are forced to move by the ministry of magic, put into protective custody by the ministry. However, the ministry fell only a few weeks later, and there was evidence that it couldn't be trusted, even then. Are they still alive, and if so, how did they survive the purge from the ministry?
According to the book, the Order of Phoenix offered their protection to the Dursleys:
Harry Potter: "The Order is sure Voldemort will target you, whether to torture you to try and find out where I am, or because he thinks by holding you hostage I'd come and try to rescue you. <...> You've got to go into hiding and the Order wants to help. You're being offered serious protection, the best there is." - Harry Potter and the Deathly Hallows, chapter 1
They suspected that the Ministry of Magic has been infiltrated:
"I thought there was a Ministry of Magic?" asked Vernon Dursley abruptly. "There is," said Harry, surprised. "Well, then, why can't they protect us? It seems to me that, as innocent victims, guilty of nothing more than harboring a marked man, we ought to qualify for government protection!" Harry laughed; he could not help himself. It was so very typical of his uncle to put his hopes in the establishment, even within this world that he despised and mistrusted. "You heard what Mr. Weasley and Kingsley said," Harry replied. "We think the Ministry has been infiltrated." - Harry Potter and the Deathly Hallows, chapter 1
As far as I know, there is no information about the life of Dursleys during the year preceding the fall of Voldemort. Talking about their future, J.K.Rowling has only confirmed the following:
"Harry and Dudley would still see each other enough to be on Christmas-card terms, but they would visit more out of a sense of duty and sit in silence so that their children could see their cousins." - Open Book Tour, Morning Appearance, NYC, October 19, 2007
Hence, at least Dudley Dursley is alive. I assume that other Dursleys have also survived. After Voldemort died or a short time thereafter, they would be able to return home.
"Protective Custody" typically means they've shunted you into something like the Witness Protection Program, and your identity is secret even from most members of the government. For the family of The Boy Who Lived, I'm sure their new names and location were known only by a select few (or possibly only one). Thus, unless one of those few were captured or turncoats, the Dursleys would have been safe.
It's also highly unlikely that the big V's people would spend a lot of time going after them - it's no secret that there's little love lost between them (unlike with Hermione's family) and Voldemort (if I recall) consistently treats Harry like an orphan.
As there was little love lost, I'm not sure what finally happened with them, but I'd bet it was one of two things:
a. Harry let them know it was over, they went back to their normal lives
b. They're living on in their new lives, as a normal family.
In The Cursed Child we see Dudley returning Harry's blanket he was wrapped in when left at the Dursleys front steps.
This . . . is the last thing I had from my mum. The only thing. I was given to the Dursleys wrapped in it. I thought it had gone forever and then, when your great-aunt Petunia died, hidden amongst her possessions, surprisingly, Dudley found this and he kindly sent it on to me, and ever since then — well, anytime I’ve wanted luck I’ve found it and just tried to hold it and I wondered if you . . .
So we know by the time that Albus is a 3rd year Petunia had died, and that Dudley was still in contact with Harry, on a relatively nice basis.