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Planets have migrated throughout the history of the solar system, and some planets could even have swapped positions. Assuming the Oort cloud does exist, has it always been at its current distance from the Sun? Or did it used to be closer? Or further away?

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The Oort cloud is more of a theoretical construct. It's a catch all, in effect, for small icy fragments anywhere from 2000 AU to over 100,000 AU.

From NASA:

The Oort Cloud lies far beyond Pluto and the most distant edges of the Kuiper Belt. While the planets of our solar system orbit in a flat plane, the Oort Cloud is believed to be a giant spherical shell surrounding the Sun, planets and Kuiper Belt Objects. It's like a big, thick bubble around our solar system, made of icy, comet-like objects. The Oort Cloud's icy bodies can be as large as mountains — and sometimes larger.

The inner edge of the Oort Cloud, however, is thought to be located between 2,000 and 5,000 AU from the Sun, with the outer edge being located somewhere between 10,000 and 100,000 AU from the Sun.

While the inner edge of such a cloud may have been closer or further in the distant past, as it has been disturbed by the planets, the idea of that region of space having icy planetesimals that may become comets doesn't really have any physical constraints other than those that Opik-Oort decided upon, unlike the Kuiper Belt, which does have measured boundaries.

Image of Oort Cloud from study.com:

enter image description here

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