I think the reason for Mughal military success being seemingly more rapid and substantial as compared to other Muslim dynasties can be summed in the following points :
POLITICAL SITUATION
The founder of the Mughal dynasty, Zahir-ud-din Mohammed Babur, was the eldest of Umar Sheikh Mirza, who was governor of Ferghana, which is a region in eastern Uzbekistan. Babur was by lineage the great-great grandson of Timur. Babur's early military career was full of frustrations. Born in 1483, he had assumed the throne of his father at age 12, in the year 1494. He conquered Samarkand two years later, only to lose Fergana soon after. In his attempts to reconquer Fergana, he lost control of Samarkand. In 1501, his attempt to recapture both the regions failed when Muhammad Shaybani Khan the founder of the Shaybanid dynasty, defeated him. He conquered Kabul, in 1504, after having being driven away from his patrimony and homeland. He formed an alliance with the Safavid Shah Ismail I, to take parts of Turkestan as well as Samarkand itself only to lose them again to the Shaybanids.
Hence, by the time when Babur first contemplated the idea of invading India he had already conquered Kabul. He had decided to give up on the dreams of taking back Ferghana and Samarkand and set his eyes on North India, in hopes of one day returning to take back his patrimony. At the time he had only thought of conquering the Punjab region. A task he accomplished in his second campaign in 1525, after a short campaign in 1519. Thus, at this juncture, we the political situation in North India was ripe for conflict and power changes. In Punjab, Babur prepared for a march towards Delhi to take it and all the realms under the rule of the Lodi Dynasty from Ibrahim Lodi who was currently the sultan of the Delhi Sultanate, whose own relatives, Daulat Khan Lodi and Alauddin had invited Babur to invade the Delhi Sultanate. Under the Lodi Dynasty the Sultanate had lost most of its eastern and southern as well as western territories and Ibrahim ruled over merely the Upper Gangetic plains. Meanwhile, a third contender for power and perhaps bigger threat to Babur's rise was looming in the Rajputana, in the form of the Rajput Confederacy, which was the first of its kind since the reign of Prithviraj Chauhan. This Confederacy was formed under the auspicious leadership of Rana Sangram Singh, of House Sisodiya of Mewar which had risen in prestige and power at the cost of neighbouring Malwa and Gujurati Sultanates during the reign of Rana Sangram other wise known as Rana Sanga. Hence, this standoff between the Delhi Sultanate and the Rajput Confederacy allowed Babur to sieze the moment, first in Punjab and then at Panipat.
The following events are well known, Babur defeated the Lodis at Panipat and then faced the Rajputs at Khanwa in 1527. However after his victories at Chanderi and at Ghaghra, he soon died leaving the Empire to his son Humayun whose reign was turbulent and prospects uncertain until his son Akbar assumed the Throne.
In 1530, Humayun assumed the throne from his father at age 22, and for the next 10 years his reign was filled with mixed military successes and an overall inability to contest with the remnants of the Afghan power present in the subcontinent. Finally in 1540, at the Battle of Kannauj, Humayun was defeated by the leader of the resurgent Afghan power, Sher Khan, later titled, Sher Shah Suri. Sher Khan's victory left Humayun a Prince with no kingdom. His own brothers turned against him and he was left to wander but with his close family, and loyal followers. Being chased out from the regions of the Upper Ganga Valley, Humayun could not seek shelter in the court of his brother Kamran who ruled over Kandahar and Kabul, since he had taken that part of the Empire following its division upon the death of their father. His brothers even attempted to have him killed. It was during this period, that this inexperienced and isolated Timurid Prince showed the strength of his character. By sheer will and fortitude and physical rigour, Humayun escaped absolute destruction and sought refuge at the court of the Safavid Shah Tahmasp I, marching with a mere 40 men, his wife and her closest companions.
Here the Safavid Shah urged Humayun to convert to Shia Islam. A difficult choice for the Mughal Prince n but one he made nonetheless to acquire the aid he needed, albeit his acceptance was merely an outward facade. Once the Shah was satisfied with this display of Shi'ism, he agreed to underwrite the Prince's attempt to regain power.
At the head of fresh troops, Humayun retook Kandahar (1545) and following the break up of the Sur Empire, he retook Delhi. Although he didn't live to enjoy his gains. This was possible because the successors of Sher Shah Suri were mere shadows of the founder of the Empire. And Humayun, was able to size this political weakness and lack of capable leadership.
Jalal-ud-Din Muhammad "Akbar" was born in Amarkot in the Rajputana in 1542. He was crowned in 1556 at Kalanaur, in the Punjab. By 1567, he had assumed control of the Empire from his Regent Bairam Khan and had crushed rebellious nobles to assume full control and authority over the helm of affairs. By 1576, Akbar had taken Rajasthan, Gujurat, Bihar and Bengal. And by 1590, he had expanded into Kashmir, Lahore, Sindh, Orissa and Kabul and Baluchistan.
Akbar's seemingly unprecedented and rapid successes were a result of the political fragmentation of North India following the breakup of Afghan power at Delhi. Plus, the fragmentation of the Rajput states who could not reunite under the leadership of a prominent house such as Mewar after the fall of Rana Sanga at Khanwa. However, opportunities weren't the only reason for Mughal success. And hence, we take a look at another contributing factor for Mughal successes.
DIPLOMACY AND RELIGIOUS OUTLOOK
In the process of his conquests, Akbar had crushed Afghan power in North India and he had established cordial relations and alliances with many Rajput kingdoms, through a policy of political and religious outreach as well as display of overwhelming military force. Akbar had used the lessons learned in his childhood well, as he was born in a time when his dynasty stood at the brink of extinction, and with the exception of a single incident during the siege and storming of Chittor in 1568, he had treated his Hindu subjects, rivals, spouses and vassals with respect and consideration and was tolerant of their religious views and beliefs.
We see a clear policy emerging from the Mughals towards the Rajputs since the reign of Akbar. The first, religious tolerance and engagement at a political level, treating them as warriors and nobles on par with the Iranis or Turks in the Imperial service. The second, realising that the prestige of Mewar and the potential of the Rajputs uniting once again was an ever present threat and therefore it was better to assuage them, rather than try to oppress or subjugate them entirely as was the attempt of the Delhi Sultanate which eventually stood frail before the Rajput Confederacy. Third, following a policy of providing high posts and port folios to Rajput nobles who allied or accepted Mughal suzerainty. Fourth, matrimonial relations were never the prerequisite for such alliances as many Rajputs had previously simply accepted Mughal suzerainty and had acquired high posts for themselves.
Apart from his diplomatic and tolerant religious policies towards his Rajput allies and Hindu subjects, Akbar was also the inheritor of a tradition of accepting the followers of Shia Islam as allies and partners. He continued his outreach and cordial foreign policy towards the Safavids and also never distinguished between his nobles in the basis of religion. He was also quick to recognise the value of engaging in trade with European traders, namely the Portuguese, and both as a means of understanding Christianity (Akbar took keen interest in understanding other religions and also adopted many Hindu practices in his lifetime) and as a means of extending a cordial message to the Portuguese, which served to ensure safe waters for Indian trade.
TECHNOLOGICAL AND MILITARY ADVANTAGE
While this point is often brought up in discussions revolving around the reasons for Mughal success, there are also many problems with stressing the perceived technological advantage of the Mughals too far. By the time the Mughals, under Akbar and Bairam Khan began their second round of conquests, the Afghans had acquired gunpowder weaponry. Indeed the reason why Samrat Hemchandra, at the Second Battle of Panipat, lacked any gunpowder weaponry, or cannons, was not because of lack of these weapons but rather because Ali Quli Khan Shaibani (later Khan-i-Zaman), who had been sent ahead with a 10,000-strong cavalry force, chanced upon Hemu's artillery which was being transported under a weak guard. He was easily able to capture the entire train of artillery from the Afghans who abandoned the guns and fled without making a stand.
And back in 1556, according to ʽAbd al-Qadir Badayuni, at the Battle of Tughlaqabad, Hemchandra was able to field 500 falchonets and some cannons against the forces of the Mughals led by Tardi Beg Khan.
Therefore, while the Mughals would eventually come to dominate the North Indian battlefield owing to their artillery, it must be kept in mind that what earned Akbar his successes were not simply his cannons but also competent leadership, delegation of power to his Rajput generals especially the Rajas of Amber, and his political and strategic foresight which led him to establish a "natural frontier" for his young empire.