The composition of the eraser does cause some differences in how the graphite is removed, but in general I've found that a dabbing twisting motion is best regardless of what the eraser is made of.
Soft putty erasers I find aresoft putty erasers to be the best for soft pencils, and a hard grit eraser best for H4 and harder, but be gentle as it can shred the paper fibers.
The StaedlerStaedtler Mars Plastic I find is a good general purpose eraser, in my opinion.
And as a final tip, keep the eraser clean,: I find rubbing the built up graphite off with your thumb to be a good tipaction prior to using it.