The Mughals have a tenuous connection at best to the Mongolians, especially if we go all the way back to Genghis. The Mughal line starts with Timur. Timur also had a very tenuous link (albeit stronger than his Mughal descendants) to the Mongols. Timur's father was a noble in a group of peoples known as the "Barlas". The Barlas had originally been a Mongolian tribe that may have had connections to several other Mongolian tribes, including Genghis Khan's. Now the Barla eventually settled in Central Asia and eventually lost much of their Mongolian heritage, adopting Islam being a key change. Timur had 4 sons, his third son Miran Shah ibn Timur is the First Mughal Emperor's Great Grandfather. Thus Timur is the Great, Great, Grandfather of the First Mughal Emperor Zahir-ud-din Muhammad Babur. The Mughals would often play up this connection and the Mughal official records and any court histories would make sure to play up the connection between Barbur and Timur.
Connection to Genghis Khan is really dodgy. Barbur's mother, Qutlugh Nigar Khanum was a distant ancestor of Genghis Khan's. While Barbur and the Mughals did like to play up the connection for prestige purposes, the actual connection was pretty much nonexistent. Before they invaded India and became the "Mughal Dynasty" Barbur and his Central Asian kingdom claimed they were descendants of Chaghta, son of Genghis.
If you're interested, I would suggest reading "The Mughals of India" by Harbans Mukhia.
Zahir-ud-din Muhammad Babur commonly called Babur (first emporer of Mughal India) is a very distant descendant of Genghis Khan and also to Timur (Tamerlan). So what does that tell us? Not that much exept that there is some relation by blood.
Lets explore this a bit. So first we have to start with the splitting of the mongol empire into diffrent khanates. Most of those had some part of the stepp and some part of the 'civilised' world. Now a guy called Timur managed to set himself up as a emir to one of the khans but exept of in title he was the most importent figure. He goes down in history as one of the greates conqueres and manges to get control over large parts of central asia. He also fell air to the mongol way of war, he really wanted to conquere everything the mongol empire once encompased. However his empire split into pieces even faster then the mongol empire had.
The diffrent parts his empire, like with the mongol empire befor, went to diffrent groupes and people. Babur enters as a stage as ruler of such a splinter that was formerly part of Timurs and the Mongol empire. Now there was constant warfare going on, diffrent groupes trie to get controll over the stepp, the silk road and so on. Basiclly you see what usally happens on the stepp when strong leadership goes away, the diffrent tribes turn on each other.
He then goes on and manages to conquer much of north india and is the founder of the Mughal Empire. He did this be combining his understanding of the muslim ruling classes, the mastery of stepp warfare and clever use of the newest in weapons tech. He however does not manage to hold central asia and thus the direct connection to the stepp.
So how is he related to the mongols?
By blood (even if he probebly had much more turkish then mongolian)
Product of the post-mongol post-timur struggles
Product of the merging of central asian culture (that muslim persian mix) and stepp culture
His way of war can be seen as a modern spin on the older stepp tactics, and his organisations are those of the mongol imperial army
So now we know about the first emporer but what about the hole rest of the Mughals period. I am much weaker on this but im still going to answer. So the stepp people that came with Babur where the minority of the minority in india. He essentially took over a existing state that mostly used a minority muslim cast as burocrates. He and his decendence managed to combine these assets very well and creat
So what is the relation between Mughals and Mongols, the Mughal empire was conquered in the style and by a decendend of mongols.
If you want to learn more about later stages of the Mughal empire and how the are connected to the stepp I have to hope for somebody else to jump in and help me.