Competing to carry the bier
What is narrated in saheeh reports from the Prophet (peace and blessings of Allaah be upon him) is that it is Sunnah to follow the funeral. Al-Bukhaari and Muslim narrated in their Saheehs from Abu Hurayrah (may Allaah be pleased with him) that the Prophet (peace and blessings of Allaah be upon him) said: “The duties of a Muslim towards his fellow Muslim are five: to return his greeting, visit him when he is sick, follow his funeral, accept invitations and say ‘Yarhamuk-Allaah (May Allaah have mercy on you) when he sneezes).’”
They also narrated from Abu Hurayrah (may Allaah be pleased with him) that the Prophet (peace and blessings of Allaah be upon him) said: “Whoever attends a funeral and prays over the deceased will have one qeeraat (of reward), and whoever attends until the deceased is buried will have two qeeraats.” He was asked, ‘What are these two qeeraats?” He said, “Like two huge mountains.”
Undoubtedly whoever carries the bier or helps to carry it until the deceased is buried will attain this great reward and more, for even if there were no saheeh hadeeth concerning this matter, the general principles of sharee’ah indicate that carrying the bier of a Muslim is prescribed in Islam, because it involves treating the deceased with kindness. But if that kind treatment will lead to the problem of people pushing and shoving, as mentioned in the question, then warding off that evil takes precedence over trying to do good. The reward of the one who hopes to gain reward by carrying the bier is not greater than the reward for the one who forsakes it out of consideration towards his Muslim brothers, and not because he is incapable of doing it.