Bukhari

Hadith Commentary:
There is great virtue in initiating the greeting of salam, as Hazrat Abu Umamah radi Allahu anhu reports that the Messenger of Allah sallallahu alayhi wa sallam said:
“The person who is closest to Allah among the people is the one who initiates the greeting of salam.” (Sunan Abi Dawud, Al-Adab, Hadith: 5197)
However, it is extremely important to observe the etiquettes mentioned in the above hadith.
Yes, if people who are walking meet each other, then the one who initiates the greeting of salam among them is superior, as Hazrat Jabir radi Allahu anhu states that among those who are walking, whoever initiates the greeting of salam is superior.
(Al-Adab Al-Mufrad, Hadith: 983)
In another hadith, there is further clarification that the Messenger of Allah sallallahu alayhi wa sallam said:
“The rider should greet the one who is walking, and the one who is walking should greet the one who is sitting, and those who are fewer in number should greet those who are more in number.
Whoever replies to the greeting will receive a reward, and whoever does not reply will be deprived of the reward.”
(Musnad Ahmad: 444/3)