Hello Artem, nice to meet you.
The verb "To dip" means to fall.
So, your sentence should read something like this:
"The American immigration rate rose slightly, but soon after that, it took a dip and continued to fall."
a few grammar points:
When using subjects that are singular and specific, we use the article "the". We only the zero article with Plural and uncountable nouns.
There also has to be subject verb agreement throughout the sentence. So, if to dip means to fall then we cannot say it rose because it's contradictory to its meaning.
I hope this helps.