If the question is: How do people travel to the local shopping centre?
As an answer, can I say ´´by bus, by car, by walking´´. Are ´´by walking´´ and ´´by car´´correct? It doesn´t sound right to me. Is there a different (maybe more natural) way of saying that?
Thanks :)

Question

Answer

on foot
or
they walk

Answer

One would say 'on foot' not 'by walking.' In English there are always exceptions to rules :-)

Answer

All your examples are perfectly fine ways to answer the question, but to sound more natural, I would suggest:

They take the bus.

They drive.

They walk.