This is a benefit in regards to the one who precedes when entering a doorway or pathway. Most Muslims prefer the one on the right to precede and the majority of the Muslims are under the impression that this is something from the religion. The truth is that the one who is older, even by a day, should go first.
I asked our Shaikh about the action of the people of preferring those on the right side to enter [thorough a door etc.] before others?
The Shaikh said that he does not know anything on that subject. Then he said, if only they preferred those who are older in age [i.e. that is better] based on the hadith of the Prophet sallahu alayhi wa salam:
“Let the older one speak. Let the older one speak”.
[Taken from: Masaail Abi Umar as-Sadhaan li Imaam Abdul Aziz bin Baz pg. 28]
It is preferable for the one who is older to precede while entering. This was the practice of some of the Salaf.
مَالِكَ بْنَ مِغْوَلٍ ، يَقُولُ : مَشَيْتُ مَعَ طَلْحَةَ بْنِ مُصَرِّفٍ حَتَّى انْتَهَيْنَا إِلَى زُقَاقٍ ضَيِّقٍ ، فَتَخَلَّفْتُ وَتَقَدَّمَ طَلْحَةُ ، فَالْتَفَتَ إِلَيَّ وَقَالَ : ” لَوْ أَعْلَمُ أَنَّكَ أَكْبَرُ مِنِّي بِيَوْمٍ أَوْ لَيْلَةٍ مَا تَقَدَّمْتُ
Malik ibn Miqhwal said: I walked with Talha ibn Mussarif until we reached a narrow alley/path. So I went behind and preferred Talha to precede. So he turned to me and said: “If I knew you were older than me by one day or night, I would not precede you”
[Reported by al-Kharaaiti in Makaarim ul-Akhlaaq no. 365 and the Muhaqiq of the book, Abdullah ibn Bajaash, graded it Saheeh pg. 633]
And Allah knows best
Faisal ibn Abdul Qaadir ibn Hassan