English word for when grammatical rules causes words to change ("bend")

I'm basically looking for a translation for the Swedish word "böja" (literally "to bend") in the context of grammar.

This verb is used generally for when words change according to grammatical rules, here are some examples when I translate literally from how I would use "böja" in Swedish sentences:

In present continuous, the verb "run" bends to "running".

The genitive bending for "James" is "James's"

The plural bending for "umbrella" is "umbrellas"

Etc.

So is there such a general grammatical term in English? Even if there isn't that much use for such a word in English grammar - you don't have to deal that much with words that change according to clear rules - it ought to be practical when studying or discussing other languages, especially ones that have many cases and genders.

The term you are looking for is inflexion (also (but less properly) spelled inflection). It is derived from Latin flecto "to bend", and it indicates any kind of change to a word where it gets an ending.

Note that affixes other than endings, as in hard → hard-ship or combust → combust-ion, don't count as endings. Keep in mind, though, that there are 'endings' that don't come at the end of a word in some languages; perhaps another term would be more appropriate, but I wouldn't know which. As you can see, Wikipaedia uses the word inflexion (as a countable noun) for this as well. It's rather complicated to explain what kinds of affixes normally count as inflexion and what kinds don't.

The inflexion of a verb, or the collection of all of the inflexed forms of a verb, is also called its conjugation; of a noun/pronoun/adjective, its declension.

The collection of all inflexed forms that belong to a certain specific type of verb/noun/etc. is also called its paradigm.