Was the expectation of a Hand of the King’s family to move entirely to King’s Landing? If Rickon had been older when Ned got his job offer, would all the Starks move to KL with Ned (sans Robb who is to remain a heir of Winterfell?) Would Rickob or Bran be required to stay back with Robb? How does the whole “always a Stark in Winterfell” plays out then when Robb has to leave to war if he is the only Stark left behind?

To answer this, let’s look at what the original plan was vis-a-vis the Stark’s move:

Ned stood, and took her in his arms, and held her face close to his. “Rickon is very young,” he said gently. “He should stay here with you and Robb. The others I would take with me.”

“I could not bear it,” Catelyn said, trembling.

“You must,” he said. “Sansa must wed Joffrey, that is clear now, we must give them no grounds to suspect our devotion. And it is past time that Arya learned the ways of a southron court. In a few years she will be of an age to marry too.“

Sansa would shine in the south, Catelyn thought to herself, and the gods knew that Arya needed refinement. Reluctantly, she let go of them in her heart. But not Bran. Never Bran. "Yes,” she said, “but please, Ned, for the love you bear me, let Bran remain here at Winterfell. He is only seven.”

“I was eight when my father sent me to foster at the Eyrie,” Ned said. “Ser Rodrik tells me there is bad feeling between Robb and Prince Joffrey. That is not healthy. Bran can bridge that distance. He is a sweet boy, quick to laugh, easy to love. Let him grow up with the young princes, let him become their friend as Robert became mine. Our House will be the safer for it.”

So the original plan was that Robb would be the Stark in Winterfell, Rickon was too young to travel, and Catelyn should stay to advise Robb and mother Rickon, but that Sansa and Arya would go south to be wed and Bran would be at court as a companion to princes.