The way the "problem" has been framed there is no viable solution. There is just no way anyone can walk 400 miles carrying a child and a small dog and a heavy pack containing everything all three of them will need.
Regardless of an individual's fitness level, experience, and/or equipment, there is no way that the task being described can be accomplished. Just cannot happen. No amount of gear or advice can make it do-able either.
That suggests to me that it is time to re-evaluate the supposed problem.
"Last resort is to travel to our destination on foot." is the way the OP stated the situation, I believe...Nothing stipulated about needing to be self-contained from the get go.
Your point is quite valid - walking 400 miles would indeed be quite a saga and an enterprise that most of us, certainly me, (and I am an experienced, practiced hiker) would find daunting. But is it impossible to the extent that it should be rejected out of hand? People have walked comparable distances - consider the case of the Mormon handcart pioneers who walked from Illinois to Utah, or, indeed, the majority of the travelers along the Oregon and Santa Fe trails. Even though they did have wagons, most emigrants typically walked alongside.
It is certainly true that you can't walk self-contained for anything like 400 miles. The question is then, "How do you resupply?" The answer to that question is just a bit complex, but it is worth considering.
I am sure that most of us, including the most avid hikers, cyclists, and runners, if faced with the necessity of a 400 mile journey, would rather do it in a day by car rather than y a slower, more cumbersome means of transport. But that doesn't mean you should never consider how you might accomplish such a feat, and what you would need to accomplish it.
I wonder how far people are walking now in Syria?
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