Are you talking about writing.... as in handwriting skills, or writing as in writing ABOUT what they read?
In 1st grade, writing/handwriting and reading comprehension (ie: writing about what they read) is taught and is part of the curriculum.
My son is in 1st grade, the Teacher, sent home a cute little flier to the parents... about HOW to help their child, with reading comprehension with anything they read. ie: then along with that, it entails writing... about what they read.
Now at 1st grade, they are not going to write a summary like how a 4th grader would.
But yes, my son writes about what he read, and it often entails 3 or more sentences or a 5 line paragraph. Kids this age, don't exactly write in paragraphs, yet.
For your child in Kindergarten, my son when he was in Kinder, also got worksheets in which they'd write a few sentences. Basically all that means is telling about something. So in the case about writing about an event in their lives, the child can simply write: "I like Christmas. It was fun. I got a new puppy. He is cute."
It doesn't have to be, complex... like how a 4th grader would write.
AND perhaps, ASK the Teacher, for examples. My kids' Teachers, especially in the lower grades, sends home an example flier... to the parents. And on the students worksheets/homework, there are ALWAYS examples about the assignment.
Even if a child is reading above grade level, writing/reading comprehension... is different.
My son's 1st Grade Teacher says, simply to ask your child questions... about what they read. And make sure the child knows what verbs/adjectives/nouns are. Because, when they write about what they read, then they can construct sentences, about what they read.
With my son, the Teacher says that all the 1st graders can work on writing... as in, writing sentences about what they read. Because, at this age... they often will just make simple sentences (ie: I ran in my yard. I like recess. My Mommy cooks dinner, etc.) They are not yet, making or constructing complex thought out sentences. But they are being taught, how.
Or, I just "chat" with my son about what he is writing in his assignments. So, for example, he will make a sentence saying "I like my ball." So then I will tell him "What about the ball, how can you make your sentence more interesting??? Any adjectives or descriptive words you can use???" So then he will think about it more, and you need to cue your child and prompt them. Then my son will write "I like my bouncy ball because it is red and blue and bounces high." So now, THAT is a well written sentence. More interesting, too. That is the gist, of what my son's Teacher tries to encourage in the kids' writing.
Or if my son is reading something, I will simply ask him about it. ie: what was the main idea in this story? Who is the main character? Did the boy have a friend? What was his name? What did they do? Why?
Just interact with your kid, about what they are reading. And then about writing it down.
But remember, this is only Kindergarten or 1st grade.
The kids are practicing these skills. It is not habit yet. Nor is it intuitive yet.