Learn English – ‘run an errand’ – what does ‘run’ mean

expressions

a. He did an errand for his mother.
b. He ran an errand for his mother.

I'm not a native english speaker. This expression 'do an errand' is easy for me to understand. We also have this expression 'do an errand' in our tongue. But in 'run an errand' I don't know what does 'run' mean.
I understand what 'run' means in these sentence:

Can you run as fast as Mike?
I ran to meet her.

But when natives use 'run an errand'… What nuance of meaning of the word 'run' is here?
So, Does 'run an errand' mean 'He runs to do an errand'? If I'm walking to do an errand, it is not run an errand but do an errand? I'm confused. T.T (Oh! she is running in the picture)

enter image description here

Best Answer

"Run an errand" is an idiom. Its meaning is not built up in a natural way from the meanings of its parts. I can think of no other expression where "run" essentially just means "do", but it does here.

I think LawrenceC's answer gives a good explanation of how it makes sense for this particular idiom to have evolved.