From Julia's memoirs:
"Victor (this was a favorite pet name for the Captain, after he had read to me the Triumphs of Victor Emmanuel) returned home a few days after the commission as Colonel of the 21st Illinois was received and was soon ready to join his regiment. Strange to say, I felt no regret at his going and even suggested that our eldest son, just then eleven years old, should accompany him. I think it was a tender thought for my beloved husband that prompted this suggestion, as well as a desire to gratify the importunings and pleadings of our boy Fred. You must not forget, though, that these regiments were only called out for three months, and I considered it a pleasant summer outing for both of them. I did not think Captain Grant's occupation in Galena entirely congenial to him, and was willing, therefore, that he should go out on this expedition as he wished to, no matter how lonely I might be without him. The Colonel and Fred bade us adieu. Many friends assembled to say goodbye. Colonel Grant's regiment was at Springfield, I think.
When Colonel Grant was ordered to Missouri, he wrote me he would send Fred home from Quincy, Ill., saying, 'We may have some fighting to do, and he is too young to have exposure of camp life.' I instantly wrote to him, 'Do not send him home; Alexander was not older when he accompanied Philip. Do keep him with you.' But I was too late. Fred had started. Colonel Grant was rather amused at my letter. The dear boy demonstrated, however, that I was right, for he walked the whole distance from Dubuque to Galena, seventeen miles, and carried his own knapsack, not a light one by any means. The cars had started when he arrived, and imbued with his father's future tactics, no doubt, he pushed ahead and was quite exhausted when he arrived."
Julia seems to have none of the quibbles of a modern day parent in sending her eleven year old son with his father off to the war. Of course, not much was expected to come of it at the time, or certainly not for any great length of time, but my jaw nearly dropped to read of her telling Grant not to send Fred home comparing him to Alexander
Especially when she has compared their first foray into camp life 'a pleasant summer outing.' What Grant had seen of war, Julia had not. She could have no possible concept of what that might mean in real terms and it appears her 'romantic' streak may have overtaken her here. Grant sensibly sends his son home for now, and Fred shows all the mettle of his father in walking 17 miles to get there after missing his train. The determination of the father resides in the son.
Here is further from the link in the OP in relation to this:
"
It was also the general who urged his wife to bring their family to live with him whenever it was deemed reasonably safe and at one point his eldest son Fred, then twelve, lived with him in camp during the 1863 battle of Vicksburg. The young man teenager not only witnessed fighting but was grazed by a bullet. As word of the presence of the Grant children in camp went beyond the circle of the Union Army to the general population, each of the children also began to develop a public profile. The dark side was that it made them targets for Confederate kidnapping. One such attempted incident was prevented only when Julia’s sister Emma acted swiftly to relocate her nephew Fred from two strangers asking about his whereabouts."