In The Witch of Blackbird Pond, how does Nat and Kit's relationship change through chapters 12-14?

When Kit and Nat parted after her arrival in Wethersfield, there was a distance between them.  Nat was still upset with Kit for diving into the cold water near Saybrook.  During their last few days on the ship, Nat treated Kit with a mixture of indifference and anger.  

In Chapter 12, Kit helped Nat repair Hannah's roof.  He was surprised when she offered to assist him, but he welcomed the help.  As they worked, Nat spoke to Kit with kindness.  He asked her how her new life was.  They talked about books and the King.

When they met next in Chapter 14, Nat was upset with Kit.  She had not told him that William was courting her.  Nat confronted her about this, telling her how he found out:

"An interesting cargo we had this trip.  One item in particular.  Sixteen diamond-paned windows ordered from England by one William Ashby.  They say he's building a house for his bride. A hoity-toity young lady from Barbados, I hear, and the best is none too good for her.  No oiled paper in her windows, no indeed!"

Nat spoke rudely to her before they parted ways.  Judith questioned Kit about his rudeness.

Does Kit from "The Witch of Blackbird Pond" get married?

At the story's end, Kit and Nathaniel are reunited.  Although the story ends before they are actually wed, there is no doubt that their marriage is imminent.

Nat had asked Kit to go with him before, but Kit had demurred, and, as far as he knew when he saw her last, she was going to marry William Ashby.  Realizing that they were not right for each other, however, Kit had broken off her engagement with William, who ended up marrying Kit's cousin Judith, a much better match for him, instead.  After surviving the typhoid epidemic and being accused of witchcraft, Kit had come to accept that she would never be at home in Connceticut Colony, and "all her plans now turned toward Barbados", and Nat, whom she finally perceives as being the one who has loved her all along, just as she is.

When Nathaniel arrives and learns that Kit has not married Ashby, he knows from Kit's demeanor that she will accept his own proposal this time.  Being a proper gentleman who wants to do things right, Nathaniel resolves immediately to ask her Uncle for her hand.  Kit wants to see the new boat Nat has managed to acquire, but Nat refuses.  He wants her to wait until they are married; when he takes her aboard, "it's going to be for keeps".  The book ends with the two of them walking to the Woods's residence, to get the approval of the family and make their plans official (Chapter 21).

In The Witch of Blackbird Pond, what occurs between Kit and Nat at the end?

Well, for all you die-hard romantics out there, you are going to love it. It is somewhat predictable, but the ending of this great novel starts with Kit being left by herself with the Woods and planning her move to Barbados by herself. She realises what has been obvious to the reader all along--that she loves Nat:

It must have been Nat from the beginning, she admitted now, and with that knowledge came a sureness that she had never known in all the last bewildering year.

However, as she ponders this, and eagerly awaits the return of The Dolphin, one day she sees a "trim little ketch" tied up at the harbour. She recognises Nat and he tells her that this is his boat now, and rather ironically, he is named it after Kit herself, callling it "The Witch":

"The ketch has a mind of her own. She's contrary as a very witch herself."

Thus it is that Kit takes Nat home to speak to her uncle, Matthew Wood, and ask for Kit's hand in marriage. Kit looks forward to life as a Captain's wife but also staying with Hannah upstream and a well-deserved happy ending.

In The Witch of Blackbird Pond, what occurs between Kit and Nat at the end?

After Kit helps Hannah to Nat's ship, the people of Wethersfield burn down Hannah's house as they have judged her to be a witch. We, the reading audience know and understand that this is not true. Nat invites Kit to come and sail away with them, but Kit refuses. She believes that Mercy's sickness still requires her close attention and care. When she arrives home to check on Mercy, she ironically discovers that Mercy doesn't need her. This makes her regret having not taken Nat's offer. However, Nat continues to believe that Kit refused to accompany him because she wanted to be with William. Nat takes Hannah to live with his grandmother. Kit will continue to endure the growing scrutiny of the town.