Learn Portuguese (Portugal): A população da cidade cresce cada vez mais. - The population of the city grows more and more.

While you might occasionally hear mais e mais, cada vez mais is the most natural and standard way to express an ongoing increase in Portuguese. It literally translates to "each time more" and is the correct way to form progressive comparisons like "increasingly" or "more and more."

Does the phrase cada vez mais need to change because população is a feminine noun?

No, cada vez mais is invariable. In this sentence, it functions as an adverb modifying the verb cresce (grows). Because adverbs don't have gender or number, the phrase stays exactly the same no matter what the subject is.

The English translation says "grows", but we often say "is growing" for trends. Can I use the continuous tense here in Portuguese?

Yes, you could absolutely say está a crescer (is growing), which is the standard continuous form in European Portuguese. However, the simple present tense cresce is also very commonly used to describe ongoing general trends over time, which is why it works perfectly here.

How would I change this sentence to say the population growth is slowing down?