Hereunder are few incidents regarding those who would cry on thinking of Rasulullaah SalAllaahu Alayhi Wa Sallam after his demise:

1.) A woman once came to Aaishah RadhiAllaahu Anha and requested, “Please show me the grave of Rasulullaah SalAllaahu Alayhi Wa Sallam!”

Aaishah RadhiAllaahu Anha opened the room. When the woman saw the grave of Rasulullaah SalAllaahu Alayhi Wa Sallam, she started to weep so much that she collapsed and passed away right there.
(Shifaa vol. 2 page 23, Mawaahib vol. 3 page 595 quoting from Ibnun-Najjaar, Zarqaani vol. 12 page 196)

2.) Umar RadhiAllaahu Anhu was patrolling the streets one night when he saw a light coming from the house and he heard the voice of an old lady. She was spinning wool and singing a few couplets, which meant,

May Allaah Taála accept the salutations of the pious and selected ones upon Muhammad SalAllaahu Alayhi Wa Sallam.

O Rasulullaah SalAllaahu Alayhi Wa Sallam! You worshipped Allaah Ta’ala each night and you wept before the dawning of each day.

I wish I knew whether I would be able to join my beloved

Because death comes in different ways and I know not how I shall die

Ah! Will I ever meet with Rasulullaah SalAllaahu Alayhi Wa Sallam afterwards
or not? 

When he heard these couplets, Umar RadhiAllaahu Anhu sat down weeping in love and memory of Rasulullaah SalAllaahu Alayhi Wa Sallam.

(Zuhd war Riqaaq of Ibnul Mubaarak no. 1024)

3.) Umar RadhiAllaahu Anhu once visited Jaabiyah (a place in Shaam) during the period of his Khilaafat. On his arrival, the people approached him and asked him if he would request Bilaal RadhiAllaahu Anhu, who was residing in Shaam, to one day call out the adhaan for them. Umar RadhiAllaahu Anhu put their request to Bilaal RadhiAllaahu Anhu who agreed to one day call out the adhaan. The momentous day finally dawned and as the people heard the adhaan of Bilaal RadhiAllaahu Anhu they were reminded of the blessed era of Rasulullaah SalAllaahu Alayhi Wa Sallam and immediately began to weep.

Such was their weeping out of the sadness and grief of being separated from their beloved Rasulullaah SalAllaahu Alayhi Wa Sallam that Aslam Rahimahullaah , the freed slave of Umar RadhiAllaahu Anhu, said, “I never saw people crying more than they did on that day.”

(As-Sunanul-Kubra of Bayhaqi no. 1974, Siyar A‘laam min Nubalaa vol 1 page 357)

4.) After Bilaal RadhiAllaahu Anhu settled in Daariyyaa (a place in Shaam), he one night saw a dream in which Rasulullaah SalAllaahu Alayhi Wa Sallam said to him, “O Bilaal! What is this separation? Is it not time for you to visit me?”

When Bilaal RadhiAllaahu Anhu awoke from the dream, he was filled with such grief and concern that he immediately mounted his conveyance and set out in the direction of Madinah Munawwarah. When he arrived before the blessed grave of Rasulullaah SalAllaahu Alayhi Wa Sallam, he was overcome by his love for Rasulullaah SalAllaahu Alayhi Wa Sallam and grief over his separation and thus began to weep excessively.

Hasan RadhiAllaahu Anhu and Husain RadhiAllaahu Anhu thereafter came to meet him. On seeing them, Bilaal RadhiAllaahu Anhu began to embrace and kiss them out of the love he had for the family of Rasulullaah SalAllaahu Alayhi Wa Sallam. They thereafter said to Bilaal RadhiAllaahu Anhu, “We desire to hear you call out the adhaan as you would call it for Rasulullaah SalAllaahu Alayhi Wa Sallam.”

Bilaal RadhiAllaahu Anhu thus ascended to a raised place in order to call out the adhaan. As Bilaal RadhiAllaahu Anhu commenced the adhaan and called out “Allaahu Akbar Allaahu Akbar!” Madinah Munawwarah began to echo with the blessed voice of Bilaal RadhiAllaahu Anhu. As he continued and called out the next words of the adhaan, the echoes in Madinah Munawwarah increased, reaching the people, until the women and children emerged from their homes, crying out of the deep remembrance of the blessed era of Rasulullaah SalAllaahu Alayhi Wa Sallam.

Such was the love they had for Rasulullaah SalAllaahu Alayhi Wa Sallam that never before had there been a day after the demise of Rasulullaah SalAllaahu Alayhi Wa Sallam in which the people of Madinah Munawwarah cried more profusely than that day.
(Tareekh Dimishq vol. 7 page 137, At-Targheeb Wat-Tarheeb vol. 2 page 235, Siyar A‘laam min Nubalaa vol. 1 page 358)

Mufti Taqi Uthmani D.B has written, “The natural demand of love is that one has love for any such thing which is connected to the beloved, especially his house, his city, the alleys of his city, as well as the doors and walls. In short, everything is beloved. The natural and binding demand of the love of Rasulullaah SalAllaahu Alayhi Wa Sallam is love of Madinah Tayyibah. For this reason, Allamah Buseeri’s Qasidah Burdah commences with mention of Dhu-Salam - an area close to Madinah Tayyibah.”
(Na’tun Nabi page 178)