https://cvc.cervantes.es/trujaman/anteriores/septiembre_02/1...

I am not sure if this is an accurate translation, but the phrase sounds more natural in Spanish with the adjetives located after the nouns.

Hola:
Pues opino que depende un poco del contexto. En los dos ejemplos que incluyo a continuación, yo lo traduciría como “A veces, las pequeñas acciones son las que cambian el mundo/generan grandes resultados/generan grandes cambios”.

The Little Things Affect the Big Things
I always say, ’Don’t think one person can’t make a difference,’” says Kristie Disbrow, founder of Matthew’s Ministry, a non-profit organization that provides weekend food service to schoolchildren every week. “I think sometimes people see a really big problem and think, ‘I’m not getting involved. What could I do?’ The answer is, a lot. When you take a step of faith and get started, everything else falls into place.
https://www.wilsonart.com/case-study/little-things-affect-bi...

Little Things Affect Big Things: Diaper and Period Banks!
https://www.facebook.com/events/dekoven-house-community-cent...

I agree with most of my colleagues here, since their sentences for make sense to me. However, this sentence is one I've seen, read and heard in many contexts with a shorter version of what Analía Quinlián proposes. That is the one I've written above. It might depend on where does the spanish speaker come from, but I strongly believe that in Spain this sentence has been mentioned more than once and has a strong message behind it, such as the original string.
The literal translation would be "Sometimes, it's the little things the ones that make a difference." This sentence can be also found in the English language, and would reflect the closest meaning of the sentence: a little change can scale up and become a very important change that, eventually, will make a difference and change big things on its way.
Thank you very much to all users for responding and collaborating to clarify this great motto.