Why did the Lord of Winterfell execute the deserter of the Night's Watch

Question

In the very beginning of Game of Thrones, we see a Night's Watch soldier, Gared, deserting his brother after seeing the White Walkers and Wights. He is later captured by Stark soldiers and brought before Ned Stark, the Lord of Winterfell. Ned Stark executes him in the name of King Robert.

The crime for deserting the Night's Watch is execution as stated later by Samwell to Jon Snow. But why did a Lord of Winterfell execute a deserter of the Night's Watch. Why didn't Ned hand him over to Lord Commander Mormont?

In season 2, when Theon is holding Winterfell and Bolton men are at the gates, Maester Luwin tells Theon to run to the north and join the Night's Watch. He says that, once he joins the Night's Watch he will be out of reach of any Lord or King of Westeros.

So, Ned being honorable and all should have sent Jared back to the Wall for Commander Mormont to decide what to do with him no? Mormont even tells Jon that even though they take all kinds of oaths, he still forgives brothers who go to Mole's town brothels.

Answer 1

Executing deserters of Nightwatch is the law in the entire Westeros, it just happens that most of deserters naturally will be caught in the north. So it is not the case that deserters should be brought to either Lord Stark or Lord Commander. Most likely any soldiers that catch one will execute him on the spot. It is just how Ned Stark rolls so when his soldiers caught the deserter, they are obviously already under orders that executing is to be done by Lord Ned himself. There is no indication that other Lords of North do it the same way, and actually most likely they don't. Law of Seven Kingdoms says deserters are punished by death and that goes for everyone, not just Nightwatch, so it is not Lord Commanders responsibility, but of every Lord and any man for that matter.

Furthermore, Nightwatch deserters are taken to be the most dangerous kind of outlaws. He who decides to desert already know that he is already on death penalty, so other moral/ethical limits usually falls down quickly. Worst crimes are being done since he is already dead if caught. It is mentioned in the show and we can see that when Bran is attacked and meets Osha, and it is more spoken of in the books. Therefore, soldiers finding one will not likely end up in transportation of any kind, exception being overly honorable Lord Stark. Deserter in question is neither killer nor rapist and probably would not commit any atrocities on his run, but that does not influence death penalty for desertion.

Answer 2

He had two choices. Execute him there, or take him at least a weeks journey away to be executed. Since he knows the outcome will be the same, execution there makes more sense. Less waste of resources (time, food, etc.) and less danger to themselves.

Further, it is implied (in a conversation with his son where he says: "He who says the word, swings the sword" (thanks: @bestguess)) that he thinks bringing him to be executed is less ethical than doing it himself.

Answer 3

Desertion is a crime and if a deserter enters a lords territory he should execute them.

Within the ranks of the Night's Watch, desertion is considered a form of oathbreaking. Deserters are not only hunted down by sworn brothers of the Night's Watch but also by lords should a deserter enter his lands.

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_{Desertion - Game of Thrones Wiki}

There would be no need for a lord (Ned) to bring a deserter back to the wall, it would just waste time.