Remembering the great martyrdom of Imam Hussain (AS) makes us more near to the Islamic teachings for which Imam Hussaun (AS) has sacrificed his life. Carrying the bier of Imam Hussain (AS) reminds us to be better human beings. Hajj rituals remind us about the sacristy of Prophet Ibrahim and Prophet Ismael (AS).
Prophet Muhammad (SAWA) wept on his uncle Hamza knowing that martyrs are alive, but we express our grief on the martyrdom of the most beloved servants of Allah (SWT).
Wassalam.