I assume the questioner would easily be satisfied if we can share two aspects, one the historic lineage of Vedas and its branches. Second, the evolution of theory -- to the concept -- to practical applications. I will answer the first part, for the second part, I will copy past an amazing post written by Ram Abloh (Amazing Vedic Scholar) in Quora. I will also post its link.
Part 1: (Kindly bear with the complexity as this is a complex question).
We have to understand that there were over 1180 Shakas of Vedic studies and implementation. Today we have only 14 Shakas out of which 6 are in practical implementation (continued oral lineage, Gov. of India has created a portal and has invited these families so that they can record their recitation). If we do the math, less than 2% survived after thousand years of invasions, displacements of families, loss of soil fertility due to drought, flooding, death of cattle, death of family lineage. Why is this important? It's important because the readers have to realize that each Vedic shaka (branch) has its own Upanishad. This Upanishad is the philosophical concept that surrounds the mantras laid out in the Samhita section of that school. They also emulate the implementation of these mantras with their respective stories in the Brahmana/Aranyaka section of this school. Readers also have to recognize that Upanishad is not necessarily a separate book, for example: Taittiriya Upanishad is actually the 7th to 9th chapter of Black Yajur's Taittririya School's Aranyaka. Similarly, Aitareya Upanishad belongs to Rig Veda's Shakala School's Aranyaka (4 to 6th chapter). The same goes for BhrA Upanishad belongs to White Yajur's Madhyandina School's 14th Brahmana (3rd to 8th chapter). We can go on like this for many other Upanishads. By this time the questioner should realize that Upanishad is not a separate book composed at a later duration of time. Yet, one has to also ask "Does this rule apply to all Upanishads?". Let's answer that with the following section.
"Does these Upanisads have linkage back to Samhitas + Brahmanas + Aranyakas?"
As we have seen above, there are various Shakas (schools), for example, the Kathak School of Black Yajur is lost forever, only a small portion of Aranyaka and its full Upanishad survived. So, what does that mean? It means, the linkage of this Upanishad back to its Vedic mantras in the Samhitas, plus the stories in the Brahmana section and the Aranyakas are now lost forever. This School of Katha predates Taittririya School. Hence, due to the loss of its parent Samhitas, many Upanishads look as if they are separate books. But this is not a blanket statement. Many later Upanishads were written to summarize various sections of Vedic Samhitas and their Primary Upanishads by various sects. Rudropanishad is a clear example that summarizes many aspects of Black and White Yajur and Shvetashvatara. Same goes to many Minor Upanishads which compiled by various sects at a later time. The questioner should also keep in mind that, over 1000 years there are statements added or edited to the original content, for example: the vedic heritage portal of Gov. of India has recoded two recessions of the same Narayanopanishad, which belongs to Black Yajur's Taittririya School's Aranyaka. If you open this literature you will see two Chapter 10s (Chapter 10A at page 242 and at 10B 337). Now, if you go to the Archive recordings you will find two recordings for both chapters. So later sectarians eliminated content to suit their preferred diety. Many hymns of Rudra got elimited from Narayopanishad belonging to Taittririya School in the first chapter. Many can arug that it's added but the linkage back to Samhitas and Brahmanas are the proof to any content laid in the Upanishad.
Didn't our great Rishis foresee this danger? the loss of Shakas? Yes, a child prodigy at a very young age saw this. This young boy, walked the entire land of Bharat on foot multiple times and re-organized the entire Vedic essence. He collected various Upanishads (Veda+anta = Vedantas) belonging to various Shakas/Schools. Then he added Brahma Sutras and extracted Gita out of Mahabharata Itihasa. Then He took content out of Tantras and created various hymns like Soundarya Lahiri and gave a thesis called Prapancha-Sara-Tantra (The transcendental (Tantric) essence (Sara) of the 5 elemental creation(prapancha)). With this, this young boy before the age of 30 established 4 schools surrounding Bharat. He foresaw the loss of various Vedic Schools and gave this massive collection of literature that is extracted out of Vedic essence. 1000 years later many sectarian acharyas emerged and created their own subset schools specific to the diety (among the 5 major Dieites) of their preference.
PART 2: (content from Ram Abloh. Link).
The Upanishads that have a tradition of being an integral part of a particular Veda are only few in number, as some answers have said.
I had written an answer listing the major Upanishads and their location in the Vedas:
ईश केन कठा प्रश्न मुण्ड माण्डूक्य तैत्तिरी ।
ऐतरेयं च छान्दोग्यं बृहदारण्यकं दशम् ॥
“Isha kena kaThA prashna muNDa mANDUkya taittirI;
aitareyam ca chAndogyam bRhadAraNyakam dasham.”
These are the 10 Upanishads that actually form integral parts of Vedic texts (i.e. samhitA, brAhmaNam or AraNyakam):
Isha — shukla yajur veda (40th chapter of vAjasaneyi samhitA)
kena — sAma veda (last section of talavakAra brAhmaNam)
kaThA — kRShNa yajur veda (ending chapters of kAThaka brAhmaNam)
prashna — atharva veda (paippalAda samhitA)
muNDaka — atharva veda (uncertain of shAkhA)
mANDUkya — atharva veda (uncertain of shAkhA)
taittirIya — kRShNa yajur veda (chapters 7–9 of taittirIya AraNyakam)
aitareya — Rg veda (chapters 4–6 of aitareya AraNyakam)
chAndogya — sAma veda (AraNyakam of tANDya brAhmaNam)
bRhadAraNyaka — shukla yajur veda (AraNyakam of shatapatha brAhmaNam)
Other than these, two more are considered ancient and authoritative:
kauShItaki — Rg veda (embedded in the kauShItaki brAhmaNam)
shvetAshvatara — associated with kRShNa yajur veda but uncertain of textual position
So you see, most of these Upanishads are prose texts that naturally form part of the other Vedic prose texts, the brAhmaNas and AraNyakas. These prose texts have always been considered as the first commentaries on the Veda samhitAs (i.e. the metrical verses). The prose texts form an integral part of the definition of a Veda.
The word “upaniShat” itself means a secret or hidden meaning. This would not make sense unless there is something else that the Upanishad is referring to. In other words, what is the Upanishad an inner meaning of? The answer is: Veda samhitA mantras. The existence of the Upanishad implies the pre-existence of the Veda samhitA, of which the Upanishad is said to be the hidden meaning. So the Upanishad is an integral part of the Veda to fully understand the meaning of the Veda.
It cannot be said that the Upanishad came later than the Veda samhitA. The assumption is that the “original” came before the “commentary” as is usually seen in the world. However, if you look at the Krishna Yajur Veda, the Taittiriya samhitA is already a mixture of verse and prose. So the “original” and the “commentary” go hand-in-hand like a convenient guide book. So the discussions recorded in the Upanishads were most probably happening at the same time that the rishis were composing the mantras that went into the Veda samhitAs. This is why Taittiriya Upanishad says, “eShA vedopaniShat (एषा वेदोपनिषत्)” — “This is the secret meaning of the Veda”.
The arrangement into separate parts is only for the convenience of study and application.
To give a modern analogy, the Veda samhitA is like the chemistry textbook, with the theoretical expositions and the chemical equations. The commentaries (brAhmaNam, AraNyakam, upaniShat) are like notes and special instructions given by the professor only in the chemistry lab when doing experiments. Without reading the textbook, a student cannot go and do experiments in the lab. But, without the special instructions in the lab to do the experiments, the student will not understand the theories in the textbook. Both of them go hand-in-hand.
Similarly, Veda samhitA and Upanishad go hand-in-hand.
