Is there a word for 'to watch something change'? Specifically, observing something that is in the process of changing

Monitor is the word you are looking for.

verb
1. observe and check the progress or quality of (something) over a period of time; keep under systematic review
_{Oxford Dictionaries}

It would be 'to track'.

According to Merriam-Webster's Dictionary:

Track

Transitive verb

1a: to follow the tracks or trace of : TRAIL

b: to search for by following evidence until foundtrack down the source

2a: to follow by vestiges : TRACE

b: to observe or plot the moving path of (something, such as a spacecraft or missile) often instrumentally.

The simple and quite versatile study would fit quite nicely here, at least that is what I would have chosen if I were to write a biology essay on butterfly metamorphosis.

From OED v4:

study, v.

10. a. To examine in detail, seek to become minutely acquainted with or to understand (a phenomenon, a state of circumstances, a series of events, a person's character, etc.); to investigate (a problem).

b. To scrutinize (a visible object) in order to ascertain its nature or to be familiar with or interpret its appearance; loosely, to look at as if examining minutely.

This is certainly a less precise choice of word than monitor, it is after all the tenth of fifteen meanings listed in total, but I feel monitor can have some unwanted connotations. Usually the word is used when you are controlling a process. That is, keeping an eye on it in case something undesired were to happen, and in that case possibly rectify it or take some other appropriate action. It does not fit that well the act of observing and dutifully logging a natural process with the intention of acquiring knowledge.

I don't think there's a single word that specifically means to observe over time looking for changes, because the the fact that observation is being done over time implies that changes are possible, otherwise the observation over time would not be needed.

I think the previously suggested monitor is closest, but in a situation where that would be too formal, contemplate could be used. From Merriam-Webster...

Contemplate

transitive verb

1 : to view or consider with continued attention : meditate on

In this case contemplate means to pay continuous attention to a subject over a period of time. The word can mean to consider a decision, but that is not the only meaning.

"James contemplated the slow emergence of the butterfly from its chrysalis."