Children need compassion to grow psychologically fit, the children who are brought up without compassion are likely to be more aggressive and lacks self-confidence which affects so much in the growth of their personality. Moreover, when a child is given love and care he would be ready to learn and be righteous. Children are gifts from Allah and what is a gift should be paid the utmost care.
Allah says in the Holy Quran what means:
 

Allah's is the kingdom of the heavens and the earth; He creates what He pleases; He grants to whom He pleases daughters and grants to whom He pleases sons. (42:49)

Being so overwhelmed by his overflowing compassion, the prophet stressed the concept of love towards children.
  His attitude towards children was always compassionate and merciful.   He played with them and showed his adherents how they would behave with children.
Though the great responsibly he had towards his nation, his responsibility towards cementing the foundation of the Islamic state and the enemies beleaguering the new religion, he never neglected his role towards the children who will be the blocks of the social structure of the Islamic country, the following hadith stresses this concept towards children
(1)    Al-Aqra’ bin Habis was a leader of his clan. One day he came to the Prophet (s.a.w.) and saw that he was kissing his grandson Hasan (r.a.). He said that

“I have 10 sons but I have never kissed them”.

The Holy Prophet said

“He who does not show kindness does not get any affection from Allah Ta’ala”.

His mercy to his daughter's children
 The prophet's life in each

minute details is an exemplary to follow, This is a strict order to be followed, God says in the Holy Quran what mean:
Certainly you have in the Messenger of Allah an excellent exemplar for him who hopes in Allah and the latter day and remembers Allah much. (33: 21)
The prophet was keen to show his mercy towards the children in his family to show the others that they should follow his exemplary.
Anas ibn Malik (may Allah be pleased with him), the servant of the Prophet, says:

I never saw anyone who was more compassionate towards children than Allah’s Messenger (peace and blessings be upon him). His son Ibrahim was in the care of a wet nurse in the hills around Madinah. He would go there, and we would go with him, and he would enter the house, pick up his son and kiss him, then come back. (Muslim)

Nothing would occupy the prophet's heart than being in prayer and reciting Quran. The time for prayer would arrive while Amama, his grand daughter,  would be in his arms. He would then put her over his shoulders while he was praying, when he would bend down in prayer he would gently put her onto the ground, and when he would finish the prostration he would again put her on his shoulder.

When any of his grandson jumped on his shoulder while he was in prayer, he would lengthen his prostration lest the child be disturbed.
  When  the prophet ended his prayer, he would apologize to his companions saying ,

“My son had climbed my neck. I felt that he should satisfy himself. I got up from the prostration when he was satisfied”.

His mercy to other children
It should be noted that his mercy was not only restricted to his grandsons or such mercy was something special for him, the prophet was exemplary who should be followed by his adherents when dealing with their children, the prophet carried the same compassion to the others' children.

Like a merciful father, the heavy-loaded with responsibilities prophet used to share the children in their play. When the emigrants came back from Abyssinia, he race to greet them, hug their children, play with them and have fun with them.
It was his practice to give lifts on his camel to children when he returned from journeys. He was in the habit of giving children output fruits on harvesting.

Sometimes he would play with the children. Sometimes, when the children were with him, he would ask them to stand in a row facing him while he would take a seated position a short distance away, spreading his arms. He would then tell them to run towards him and whosoever reaches him first would get a prize. The children would run with full force and would not be able to stop in time: one would fall on his chest while the other on his stomach. He would hug them and shower them with affections.

 While leading peoples in prayers in the mosque, the prophet would shorten the prayer for a baby or a child was crying giving mercy to the children and their mothers.
The Prophet said,

“(It happens that) I start the prayer intending to prolong it, but on hearing the cries of a child, I shorten the prayer because I know that the cries of the child will incite its mother’s passions.” (Al-Bukhari)

 The prophet was also against giving preference to one child over the other, this would produce rancor between children and enmity, the prophet says :

“Fear Allah and treat your children [small or grown] fairly (with equal justice).” (Al-Bukhari and Muslim)

Also, the prophet rebuked one of the Bedouin because he did not kiss his children saying to him" whoever doesn't have mercy towards people won't find mercy from God likewise.