Did the Buddha's teachings change over time during his life?

The Buddha's core teaechings never changed over his lifetime.

His first teaching was about the Four Noble Truths, which includes the Noble Eightfold Path. His second teaching a few weeks later was about the Three Characteristics, namely, impermanence, unpleasurableness & not-self.

His last teachings (refer to DN 16) included how Saints could only be found where the Noble Eightfold Path was found. His final words were about impermanence.

Therefore, his teachings never changed.

At times he taught in more detail, such as explaining the Four Noble Truths as Dependent Origination (refer to AN 3.61). Or at times he taught more comprehensively, such as teaching not-self (anatta) as emptiness (sunnata).

But his teachings never changed because Buddha was Fully Enlightened before he ever taught.

Yes. The Buddha adapted his teaching according to circumstance. Here it was a matter of life and death, and the Buddha taught something different.

SN54.9:4.2: “Ānanda, why does the mendicant Saṅgha seem so diminished?”
SN54.9:4.3: Ānanda told the Buddha all that had happened, and said,
SN54.9:4.6: “Sir, please explain another way for the mendicant Saṅgha to get enlightened.”

Be sure to read the full sutta. It points out some of the dangers of misunderstanding the teachings.

I think he famously added to the Vinaya over time.

For example, he gave the explicit rule against alcohol when a monk came back drunk.

There are even a few rules added in the Maha-parinibbana sutta.

The Buddha also often adapted what he taught according to his audience's needs.

"Kesi, I train a tamable person [sometimes] with gentleness, [sometimes] with harshness, [sometimes] with both gentleness & harshness.

"In using gentleness, [I teach:] 'Such is good bodily conduct. Such is the result of good bodily conduct. Such is good verbal conduct. Such is the result of good verbal conduct. Such is good mental conduct. Such is the result of good mental conduct. Such are the devas. Such are human beings.'

"In using harshness, [I teach:] 'Such is bodily misconduct. Such is the result of bodily misconduct. Such is verbal misconduct. Such is the result of verbal misconduct. Such is mental misconduct. Such is the result of mental misconduct. Such is hell. Such is the animal womb. Such the realm of the hungry shades.'

"In using gentleness & harshness, [I teach:] 'Such is good bodily conduct. Such is the result of good bodily conduct. Such is bodily misconduct. Such is the result of bodily misconduct. Such is good verbal conduct. Such is the result of good verbal conduct. Such is verbal misconduct. Such is the result of verbal misconduct. Such is good mental conduct. Such is the result of good mental conduct. Such is mental misconduct. Such is the result of mental misconduct. Such are the devas. Such are human beings. Such is hell. Such is the animal womb. Such the realm of the hungry shades.'"

"And if a tamable person doesn't submit either to a mild training or to a harsh training or to a mild & harsh training, what do you do?"

"If a tamable person doesn't submit either to a mild training or to a harsh training or to a mild & harsh training, then I kill him, Kesi."

"But it's not proper for our Blessed One to take life! And yet the Blessed One just said, 'I kill him, Kesi.'"

"It is true, Kesi, that it's not proper for a Tathagata to take life. But if a tamable person doesn't submit either to a mild training or to a harsh training or to a mild & harsh training, then the Tathagata doesn't regard him as being worth speaking to or admonishing. His knowledgeable fellows in the holy life don't regard him as being worth speaking to or admonishing. This is what it means to be totally destroyed in the Doctrine & Discipline, when the Tathagata doesn't regard one as being worth speaking to or admonishing, and one's knowledgeable fellows in the holy life don't regard one as being worth speaking to or admonishing."

Kesi Sutta