There is no validity to the claim that immigrants cause crime.

Actually, there is, because you've misunderstood what I was saying. We're kinda on the same page though. Immigrants WILL bring crime, because they're just as likely as any other group of people to commit them. Now, the question whether they bring any crime at all is a totally different one than the question asking whether they are more or less likely to commit crimes (statistically) than the native population.

Crime statistics generally show immigrants charged for crimes at a same or lower rate than non-immigrants.

Which crime statistic? Immigration is a complex topic and the impact on crime statistics hugely depends on who immigrates into which country and who lives there.

If you ask left-leaning people to look at the historical immigration of European settlers to North America and the native population there, people will quickly say "Oh yeah, they were horrible! They came, raped, robbed and murdered the native population!", but if you ask them how they see modern immigration, they will magically say that it's all good and that all immigrants are good people and enrich the culture. Yet, those European settlers were immigrants as well (...). There is a huge bias towards how immigration is being seen there and it has something to do with the ideology you're being taught while you grow up. As I have said, the truth is much more complex and harder to grasp. And you really have to look at the statistics, because some biasses turn out to be true - and others do not.

Some groups of immigrants may have a lower chance of committing crimes, others may have a higher chance of doing so.

In Germany for example, immigrants have a significantly higher crime rate than German citizens (https://de.statista.com/themen/13642/auslaenderkriminalitaet-in-deutschland/#topicOverview) and the ratio of immigrants to citizens in criminal convictions has been rising for the last 6 years (https://de.statista.com/statistik/daten/studie/223/umfrage/verurteilte-straftaeter-nach-nationalitaet/). There are many factors to consider here, like citizens more likely to report the police to the crime of an immigrant, immigrants living in poverty or destitute conditions (making it more likely for them to steal for example), less availability of lawyers and legal assistance and so on. People from Afghanistan, Iraq and Syria were much more likely to commit violent acts - including rape - for example (https://www.bpb.de/themen/innere-sicherheit/dossier-innere-sicherheit/301624/migration-und-kriminalitaet/). While immigrants from Georgia and the African countries were more likely to commit acts of fraud or thievery.

Things might look totally different for the USA, Brazil, Mexico and so on.

It's factually true to say that immigrants always bring crime.

It's factually false to say that immigrants always bring more crime than the native population.

It's factually true to say that immigrants can commit more crime percentage-wise than the native population.

It's factually true to say that immigrants can commit less crime percentage-wise than the native population.