Mark 9.9-13, 
  Matthew 17.9-13, 
  Luke 9.36. 
In 
Probably because this vision is so open to utter misinterpretation, Jesus decided to have his kids keep it to themselves for a while, just till the context of his own resurrection helped make it make sense.
Mark 9.9-10 KWL 9 As they’re going down the hill,- Jesus commands his students
 - so no one who saw these visions would describe them
 - till the Son of Man might rise from the dead.
 10 The students kept this word to themselves—- though arguing, “What’s ‘to rise from the dead’ mean?”
 Matthew 17.9 KWL - As they’re going down the hill,
 - Jesus commands his students, saying,
 - “Nobody may speak of the vision
 - till the Son of Man might rise from the dead.”
 Luke 9.36 KWL - As the voice came,
 - the students find Jesus alone.
 - They are silent,
 - and in those days, report nothing they see to anyone.
 
Obviously they told everybody afterwards, ’cause now the story’s in 
Because the transfiguration is a vision of the End, naturally the students had the End Times on the brain. Especially since they’d just seen a major End Times figure, the eighth-century 
Malachi 4.5-6 NRSVue 5 See, I will send you the prophet Elijah before the great and terrible day of the LORD comes.6 He will turn the hearts of parents to their children and the hearts of children to their parents, so that I will not come and strike the land with a curse.
—Well, y’notice if people don’t respond to the revival they’ll be cursed, which is how ancient Christians interpreted 
Only to have the Father order them, “Listen to my Son,” the vision blink out, and Jesus back to normal. It’s not the End yet. Bit of a disappointment.
But since the topic comes up, what is Elijah’s role in the End Times?
The second coming of Elijah.
The followers of John Nelson Darby’s general view of the End Times—whom I call 
Revelation 11.3-12 NRSVue 3 “And I will grant my two witnesses authority to prophesy for one thousand two hundred sixty days, wearing sackcloth.”4 These are the two olive trees and the two lampstands that stand before the Lord of the earth.5 And if anyone wants to harm them, fire pours from their mouth and consumes their foes; anyone who wants to harm them must be killed in this manner.6 They have authority to shut the sky, so that no rain may fall during the days of their prophesying, and they have authority over the waters to turn them into blood and to strike the earth with every kind of plague, as often as they desire.7 When they have finished their testimony, the beast that comes up from the bottomless pit will wage war on them and conquer them and kill them,8 and their dead bodies will lie in the street of the great city that is prophetically called Sodom and Egypt, where also their Lord was crucified.9 For three and a half days members of the peoples and tribes and languages and nations will gaze at their dead bodies and refuse to let them be placed in a tomb,10 and the inhabitants of the earth will gloat over them and celebrate and exchange presents, because these two prophets tormented the inhabitants of the earth.11 But after the three and a half days, the breath of life from God entered the two witnesses, and they stood on their feet, and those who saw them were terrified.12 Then they heard a loud voice from heaven saying to them, “Come up here!” And they went up to heaven in a cloud while their enemies watched them.
Elijah’s known for praying for the rain to stop, and Moses (well, Aaron too) is known for smiting Egypt with lots of plagues, including the water-to-blood plague. So various Christians note the similarity to Moses and Elijah… and some Christians are pretty sure these guys are literally Moses and Elijah, even though Moses is dead. 
So again, prophets who are like Moses and Elijah. Although why these men have to be Old Testament saints, and not present-day Christians, makes no sense to me… and in any case I don’t believe these prophets are meant to represent literal men at all. But that’s a discussion for another time.
Same with Malachi. He wasn’t 
Matthew 11.14-15 NRSVue 14 “…and if you are willing to accept it, he is Elijah who is to come.15 Let anyone with ears listen!”
Gabriel had told John’s father Zechariah the same thing, even going so far as to quote Malachi.
Luke 1.17 NRSVue - “With the spirit and power of Elijah he will go before him, to turn the hearts of parents to their children 
Ml 4.6 and the disobedient to the wisdom of the righteous, to make ready a people prepared for the Lord.” 
John the baptist 
Mark 9.11-13 KWL 11 The students ask Jesus, saying this:- “Why do the scribes say Elijah must come first?”
 12 Jesus tells them, “Indeed Elijah’s coming is first;- he restores everything.
 - How is it written about the Son of Man?—
 - he might suffer much; he might be despised.
 13 But I tell you Elijah also came,- and people did to him whatever they wanted,
 - just as is written of him.”
 Matthew 17.10-13 KWL 10 The students ask Jesus, saying,- “So why do the scribes say Elijah must come first?”
 11 In reply Jesus says, “Indeed Elijah comes,- and will restore everything.
 12 And I tell you Elijah came now,- and people didn’t know him,
 - but did to him whatever they wanted.
 - The Son of Man is also about to experience the same things.”
 13 Then the students understand- Jesus speaks to them of John the baptist.
 
Of course Darbyists are more attached to their timelines than the bible, so they wanna explain Jesus away by pointing to how he said, “Elijah also came” 
But Malachi’s Elijah prophecy is ultimately fulfilled in John the baptist. It needn’t be fulfilled again. Not in our End Times prophecies, nor elsewhere.
There are always gonna be prophets who do Elijah-like things. Which stands to reason: They’re filled with the same Holy Spirit who empowered both Elijah and John. And some of them, like Elijah and John, are gonna be into long hair and leather. But the Second Coming of Elijah isn’t a literal second coming; it’s John.
And the restoration of everything doesn’t start with the End Times. It started with the birth of John the baptist. It began when John was born, grew up, proclaimed God’s kingdom, 
To listen to some of the End Times watchers, they sound like the restoration of everything doesn’t start till the rapture. So what’re they doing in the meanwhile? Well they might be spreading the kingdom, but not intentionally! They spend far more of their time worrying about it.
Don’t you worry about it. Jesus wins. Meanwhile, seek his kingdom. John the baptist is the sign the last days have begun, and God is making all things new. Join him.