Places this theme is heard in FOTR:

• TE: A lush version just after Gandalf’s cart rolls through town and we hear a Hobbit say, “It’s Gandalf”. The wagon goes through a narrow cut in a hill and we see the Hobbit holes and off in the distance, Bag End. (And just before they see preparations for The long-expected party.)
  EE: The same lush version - slightly longer - as Gandalf's cart begins to climb up a gradual slope towards Gandalf's destination. In a voice over, Bilbo says, For things are made to endure in the Shire, passing from one generation to the next. There's always been a Baggins, living here under the Hill in Bag End. (Back in his study, Bilbo puts the finishing touches to his opening chapter.) And there always will be. (see note below)
  (This early scene is considerably different in the TE & EE. The music follows a similar path for each although the scenes might be different.)

• Another perky variant just after Frodo says, “you’ve been officially labeled a disturber of the peace.” to Gandalf as they ride on the wagon.

• The perky variant moves into a lush variant as the children spot Gandalf and start running through the grass toward him.

• A quick mention of the Shire B phrase can be heard as Bilbo tells Frodo that he was "the one Baggins that showed real spirit." (FOTR EE)

• When Frodo spots Bilbo in Rivendell, the Shire A Theme plays. It moves into the Shire B Theme after Bilbo says, “Age, it seems, has finally caught up with me.” Frodo says, "I miss the Shire. I spent all my childhood pretending I was off somewhere else... off with you on one of your adventures." It moves back to the Shire A Theme after Frodo says, “My own adventure turned out to be quite different.”

• When Frodo reaches in the water to grab Sam's hand.

• During the chorus of "In Dreams". "But in dreams, I still hear your name."

Places this theme is heard in TTT:

• After Sam speculates about the stories that will told about their adventures and Frodo says, "Frodo wouldn't have gotten far without Sam." Sam tells him not to tease, but Frodo is serious. Sam considers this... "Samwise the Brave."

Places this theme is heard in ROTK:

• For Sam and Rosie's wedding. (see note below)

• When Frodo explains to Sam why he must leave on the Last Ship with Bilbo and Gandalf. First, a slight variant plays. The melody is the same but expanded - that is, phrases are repeated.

• A wider variant is heard as Frodo and Sam embrace for the last time. In this variant, the melody varies.

"There will always be a Baggins at Bag End."

Estelwyn wrote:

The same phrase of the Shire theme* which plays in FOTR EE (Concerning Hobbits scene), *exactly* as Bilbo says "For things are made to endure in the Shire, passing from one generation to the next. There's always been a Baggins living here under the Hill, in Bag End, and there always will be", ALSO plays in ROTK (Homeward Bound scene), *exactly* as we cut from the Green Dragon to Sam and Rosie's wedding kiss. In fact, the rising "swell" in the music, which underlies Bilbo's words "For things are made to endure in the Shire" exactly matches the "swell" which is heard as we see the wedding kiss.

There's also a kind of fade, or dwindling, at the end of the phrase in each case (not an exact match, as the phrase is a little longer in ROTK, but the fade is pretty darn similar). In FOTR we hear it as Bilbo wistfully says "...and there always will be.", and in the ROTK scene it is heard as the camera cuts to Frodo (just before the fade to him wandering forlornly in an empty Bag End).

This subliminal (for want of a better word) musical link between these two scenes, whether it was deliberate or not, speaks to me of both:
a) the hope, and promise of continuity, that Sam and Rosie embody (life *will* endure in the Shire, passing to the next generation etc), and
b) the bittersweet sadness of Frodo's passing, and the "ending" it signifies (there *won't* always be a Baggins under the hill, in Bag End).

For me this is just another example of the evocative power of Howard Shore's score. I am even more grateful than ever for the rich emotional subtext he has added to these films.

Then she added:

(C)onsidering these two scenes together has brought home to me both how "right" and how "wrong" Bilbo was. Things ARE made to endure in the Shire, and Bag End will pass on to the next generation, but it won't be a Baggins.

Bilbo's hesitation (or is it wistfulness?) at the end of his statement suggests that he already realises this on some level; that he already has some vague concern about Frodo's future. It is the subtlest of foreshadowings of what is to come for his dear, younger cousin. For us as the audience, seeing Frodo adrift and alone at the end of ROTK almost seems like proof that Bilbo's concern (however vague) was justified.

Sam and Rosie, on the other hand, are the living joyous proof that Bilbo's confidence in the continuity of the Shire, and life in it, was also justified. And that is an encouraging thought.

There is more in this thread worth reading. Visit the links I've provided if you wish to do so.

*This is a distinctive lush version of the Shire B Melody. Although hints and snippets and variants of the Shire B can be heard throughout the trilogy, these two iterations are the only ones that 'complete' the melodic phrase "But in dreams... I can hear you name. But in dreams... We will meet again."