Is the word moving as used here in these two sentences gerund or a present participle?

  1. The five stages of moving to a brand new place.
  2. One of the most exciting parts of moving to a new place.

Gerund are like nouns, and participles are like adjectives. You can replace moving with a noun and keep the structure of the sentence as it is:

The 5 stages of [a move] to a brand new place

One of the most exciting parts of [a move] to a new place

Only a noun-like phrase can be the object of a preposition, like of. It is therefore a gerund in both examples.

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