I'm so glad you mentioned Lot. I wanted to drag him in here but my previous answer was getting too long. (BTW I hadn't seen your post before I wrote about Noah) this thread moves way faster than I can type.
First Noah, the teachers who say he is not a picture of the Rapture are quite right, but elements of his story really do fit. And I don't mean the week before the rains, it's the pierced side of the boat (the door) the way into safety. A wooden ark, a wooden cross.
Noah is the reason I began to homeschool my kids. I heard a sermon on Noah by a homeschooler and I'd felt the Lord pulling me to do that. Me - with a husband who was dead set against it, and kids going into high school and a career in nursing. But I felt a whisper in my Spirit to go build an Ark for my kids via homeschooling and I did. God comforted me with Noah so many times when it got tough. He kept building that Ark. Laughter, mocking, discouragements galore I'm sure came his way.
Let's Loiter with Lot!!!
Those angels came to get him the night before.
They spent the night in Lot's house and finally by dawn they grab him dragging him and Mrs Lot and the 2 girls out to safety.
Backstory:
Genesis 18- Sarah giggles in disbelief when she overhears she's going to have a son, but denies it when she's found out and Abraham intercedes with the Lord over the destruction of Sodom but gives up after God agrees not to destroy it if 10 righteous are found in it.
Genesis 19 The angels arrive at Sodom and meet Lot who is sitting in the city gates (a place that means he had some respect within that city government) IN THE EVENING.
He doesn't let them spend the night in the city square. They come home to his place, and he makes supper for them.
In a foreshadowing of the Exodus, he bakes bread without leaven. An interesting detail.
After dinner, all the men of the area are outside the door demanding to have sex with the strangers in Lot's home. Lot goes out to reason with them, closing the door carefully behind him, and fails to make them understand. He even offers sex with his 2 daughters but they insist.
The angels intervene. Reaching out, they grab Lot, pull him inside and smite the Sodomites with blindness.
After that episode, they ask Lot if there's anybody else he wants to take to safety with him. He goes out and tries to talk to his daughter's fiances his future son in laws (which in Middle Eastern custom of that time were betrothed, so considered as legally married already but not yet together). They think it's a joke and laugh at him.
Obviously he takes some time to try because dawn starts to break and the angels grab him and say HURRY! They literally have to drag him, his wife and their 2 daughters by the hands.
So by the time elapsed, the angels arrived in the evening and by dawn they are out of time and drag Lot and family to safety.
As soon as they were out of the city, one of the angels told them to run for it- get to the mountains and safety. But Lot is truly related to Abraham so he bargains and asks to just go as far as Zoar
Which is why Zoar didn't burn up because the angels agreed. Which is why this video here is fascinating by Expedition Bible.
Sodom burned—Zoar did NOT: The full story of the discovery of the Cities of the Plain!
Here is the account in Genesis 19, the bits I've recapped above with some emphasis. Note the time elapsing. God doesn't begin the judgment of Sodom and Gomorrah till DAY BREAKS
So in the evening the angels come to get Lot and anyone he can persuade to come with. After dinner the Sodomites arrive and get blinded. Then Lot tries to talk his sons in law to be into coming till dawn starts. At which point the angels grab and drag him and his family by the hands. They get out of the city and Lot begins to bargain. He gets to Zoar as the sun rises and then God destroys the towns of Sodom and Gomorrah.
But before that day dawns, Lot is out trying to bring 2 people to safety who refuse to come and enter into judgment that morning. He intercedes with the angels for Zoar and Zoar survives.
Genesis 19 NIV
1
The two angels arrived at Sodom in the evening, and Lot was sitting in the gateway of the city. When he saw them, he got up to meet them and bowed down with his face to the ground. 2 “My lords,” he said, “please turn aside to your servant’s house. You can wash your feet and spend the night and then go on your way early in the morning.”
“No,” they answered, “we will spend the night in the square.”
3 But he insisted so strongly that they did go with him and entered his house.
He prepared a meal for them, baking bread without yeast, and they ate.
4 Before they had gone to bed, all the men from every part of the city of Sodom—both young and old—surrounded the house. 5 They called to Lot, “Where are the men who came to you tonight? Bring them out to us so that we can have sex with them.”
6 Lot went outside to meet them and shut the door behind him 7 and said, “No, my friends. Don’t do this wicked thing. 8 Look, I have two daughters who have never slept with a man. Let me bring them out to you, and you can do what you like with them. But don’t do anything to these men, for they have come under the protection of my roof.”
9 “Get out of our way,” they replied.
“This fellow came here as a foreigner, and now he wants to play the judge! We’ll treat you worse than them.” They kept bringing pressure on Lot and moved forward to break down the door.
10 But the men inside reached out and pulled Lot back into the house and shut the door. 11 Then they struck the men who were at the door of the house, young and old, with blindness so that they could not find the door.
12 The two men said to Lot, “
Do you have anyone else here—sons-in-law, sons or daughters, or anyone else in the city who belongs to you? Get them out of here, 13 because we are going to destroy this place. The outcry to the Lord against its people is so great that he has sent us to destroy it.”
14
So Lot went out and spoke to his sons-in-law, who were pledged to marry his daughters. He said, “Hurry and get out of this place, because the Lord is about to destroy the city!” But his sons-in-law thought he was joking.
15 With the coming of dawn, the angels urged Lot, saying, “
Hurry! Take your wife and your two daughters who are here, or you will be swept away when the city is punished.”
16
When he hesitated, the men grasped his hand and the hands of his wife and of his two daughters and led them safely out of the city, for the Lord was merciful to them.
17 As soon as they had brought them out, one of them said, “Flee for your lives! Don’t look back, and don’t stop anywhere in the plain! Flee to the mountains or you will be swept away!”
18 But Lot said to them, “No, my lords, please! 19 Your servant has found favor in your eyes, and you have shown great kindness to me in sparing my life. But I can’t flee to the mountains; this disaster will overtake me, and I’ll die.
20 Look, here is a town near enough to run to, and it is small. Let me flee to it—it is very small, isn’t it? Then my life will be spared.”
21 He said to him, “Very well, I will grant this request too; I will not overthrow the town you speak of.
22 But flee there quickly, because I cannot do anything until you reach it.” (That is why the town was called Zoar.)
23 By the time Lot reached Zoar, the sun had risen over the land.
24 Then the Lord rained down burning sulfur on Sodom and Gomorrah—from the Lord out of the heavens.
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The timing mentioned within the chapter is interesting.
Judgment is determined and set to happen in the chapter before, when Abraham and Sarah have a visit from 3 men around noon or mid afternoon- described as the heat of the day, one of whom is the Lord (pre incarnate Jesus) and the other 2 are the angels who will be sent to bring Lot out.
After they eat with Abraham they get up to leave. The angels go towards Sodom, while the Lord remains to tell Abraham what will happen to Sodom and Gomorrah. Abraham bargains with the Lord and gets to 10 righteous men before stopping.
The 2 angels arrive at evening. Lot is removed at dawn and when he reaches the safety of Zoar (NOT UNTIL) then the judgment of God falls on the cities of the plain (Sodom and Gomorrah)
There is a gap of time between the determination of judgment, the arrival of the angels to remove Lot and then Lot still has to get to Zoar. Evening to dawn to sunrise at Zoar.
Then there is the mercy of God - Lot is called righteous in the Bible, his soul vexed by the goings on around him. He is a picture of the Rapture taking imperfect believers the answer to those who like to preach a partial rapture based on works. But that is not how God works in this age of grace. Works don't save us. Trust in the Lord does. And that is how and why Lot is counted righteous. He believed God. Look at his response to the angel's message. He tried to get his son in laws to come with him, to escape the coming wrath. He believed and acted on that belief. He bargained because he believed that God was bringing judgment and that God would hear his request.
A discussion that starts up about lasagna turns into a heated debate about mozzarella. Then peripheral arguments about sauce, noodles and ricotta. Finally, both Nanas get up and each cook a lasagna and make us all judge whose is the best. 
