Hi, 

I need an advice what would be the best way to transfer visemes into Blend Shapes in Unity for the purpose of automated speech? 

Any advice is very helpful and appreciated.
Thank you

This is all pretty much automatic as long as you add the appropriate Expressions to the model, and use the AutoSetup tool to import the model in Unity. The default character should have a bunch of expression blend shapes that can be used.
There are also a bunch of ARKit visemes you can also add. Find the CC3+ Traditional Facial Profile and add it. Then, choose just to Add instead of Replace, and just the Custom Expressions. 

49% of original size (was 1024x692) - Click to enlarge

What lip sync tool are you using in Unity?

Depending if you need the Reallusion standard visemes, or the more traditional for games 15 visemes, you can switch between the viseme sets. See here:
https://manual.reallusion.com/Character-Creator-4/Content/ENU/4.0/06-Facial-Profile-Editor/8_7_and_1_1.htm