There are different explanations in the Buddhist texts on how these world systems come to cease. For example, Sūtra on the Meeting of the Father and Son states:
The time when this world system is destroyed does exist. When this world system is about to be destroyed two suns will appear in the world, and owing to the appearance of two suns, springs and small streams will dry up. Then three suns will appear in the world and owing to the appearance of three suns, large springs and great rivers will dry up. Then four suns will appear in the world, and owing to the appearance of four suns the four great rivers and the great lake Anavatapta (“that which never heats up”) into which the rivers flow will dry up and disappear owing to evaporation. Then five suns will appear in the world, and owing to the appearance of five suns the water of the great oceans will decline and disappear to a depth of one yojana.
In Asaṅga’s Yogācāra Ground there is the following: 
When not even a single sentient being remains, their material possessions too will disappear, and when the possessions cease to exist there will be no longer any rainfall. Since rain would stop falling on this great earth, all the grass, the medicinal plants, and the trees will dry up. Drought will arise and the sun disc will thoroughly scorch the entire earth.
Since there are six things that will be incinerated owing to the power of the destructive karma of sentient beings, six more suns will appear. These suns will blaze, each with four times the force of the normal sun. Then by increasing sevenfold, they will each blaze with seven times their normal force.
What are those six things that will be incinerated? The small pools and great pools will evaporate with the rise of the second sun. Small streams and great streams will evaporate with the third sun. Lake Anavatapta will evaporate with the rising of the fourth sun. The great ocean will evaporate with the rising of the fifth and the sixth suns. Because Mount Meru and the great basis are extremely stable, they will only be incinerated by the rising of the sixth or seventh suns. Then wind fans the flames of this fire, which spreads up to and including the Brahma heavens and incinerates these realms. 
Physical things will undergo three types of change: Things derived from water such as grass and so on are desiccated by the first sun itself. Things composed of water evaporate owing to the other five suns. Physical entities that are stable and hard are desiccated by two suns.
As for how the enduring world will come to burn and will be incinerated, I shall elaborate on the explanations found in the sūtra itself. It will be incinerated without smoke or without even a trace like ash left behind. In this way, as the external environment is destroyed, the world is then said to have ceased. This lasts for twenty intermediate eons, so the destruction of the world lasts twenty intermediate eons.
What is destruction by water? Initially there are seven cycles of destruction by fire. Then the water element emerges from the second meditative absorption, and this water element dissolves external world systems just as salt dissolves in water, then the water element itself disappears along with those external world systems. Such destruction also takes twenty intermediate eons.
What is destruction by wind? Initially there are seven cycles of destruction by water, then there is a single destruction by fire. Then the wind element emerges from the third meditative absorption, and external world systems disappear like a body withering because of the wind. Then the wind element disappears along with those external world systems, just as for some in whom the wind  element is disturbed the flesh shrinks to the bone. This is the destruction of world systems, and such disintegration also takes twenty intermediate eons.
According to the texts of the lower Abhidharma school, when this world system in which we reside is destroyed, the destruction takes place through fire and the sun progressively increasing in temperature. Thus in general when the external world system is destroyed it is necessarily destroyed by one of the three elements: fire, water, or wind. Furthermore, it is said that the world system is destroyed by fire seven times in succession, and then it will be destroyed by water seven times. After destruction by water seven times, the world system will be destroyed once again by fire seven times. Then it will be destroyed by wind. For example, the Great Treatise on Differentiation states:
Some say when the world system is destroyed the sun will split into seven suns, and through the force of this the world will be destroyed. Some say that at the time of the final eon this single sun itself will increase in temperature sevenfold and incinerate the world system.
In conformity with this view, the Treasury of Knowledge states:
It is destroyed by fire seven times, then by water. Thus after destruction by water seven times,  it is destroyed by fire seven times. Then it is destroyed by wind at the end of that.
Thus Treasury of Knowledge explains that after the world system is destroyed by fire fifty-six times, it will be destroyed by water seven times, making a total of sixty-three rounds of destruction. Then it will be destroyed by wind once. Asaṅga’s Facts of the Grounds, on the other hand, states that after destruction by fire and water fifty-seven times, it will be destroyed by wind.
As for the explanation in the Kālacakra texts on how the world systems come to be destroyed, this is as follows. External world systems are first generated by empty particles of space, so when they are destroyed, coarser form progressively transforms into subtler form until everything exists naturally as subtle particles of empty space. The stages of dispersal are said to correspond with the order of earth, water, fire, and wind. First, external world systems containing earth particles—that coexist through mutually cohering—separate and penetrate water, then water particles separate and penetrate fire, then fire particles separate and penetrate wind, and then wind particles separate, and like particles prior to the formation of world systems, they function in space as discrete atoms beyond the range of the sense faculties. Further, the great Stainless Light commentary states:
At the time of the destruction of this world system, “from earth” means  “water from earth” and so on, where those subtle particles of the elements that were fully conjoined disengage from the composite of subtle particles of earth and enter the composite of subtle particles of water owing to subtle particles of earth separating.
So too, those disengaging from the water element enter the composite of fire, and those disengaging from fire enter the composite of wind, and those disengaging from wind fully function in space. Thus the world system fully contracts, and some deities called fire of time who exist beneath the ground incinerate the world and turn it to ash.
In brief, there are statements in the Buddhist texts indicating there are more world systems than grains of sand in the River Ganges, and these too constantly undergo processes of formation and destruction, such that when some worlds are being formed, at the same time, other worlds are in the process of emptying. Also when the worlds undergo disintegration they are destroyed by the three elements of fire, water, and wind. Earth, water, fire, and wind themselves serially dissolve until finally they exist as empty space particles.  Āryadeva’s Four Hundred Stanzas states:
Just as one sees the end of a seed, and it has no starting point.
Similarly, Bhāvaviveka’s Lamp of Wisdom says:
As it shall be explained below in relation to the lines “With regard to all things as well, there exists no starting point,” vases and so on are states that arise in a linear stream of causation from one to the next from beginningless time. They do not exist as possessing a starting point.
As such, all things—as represented by the seed of a sprout or a vase and so on—arise in a stream of causation progressing from one cause to the next without any starting point, and one cannot identify any initial cause. So too world systems arise within a stream of formation and destruction solely owing to the power of the wheel of dependent origination as a causal process without any starting point. Moreover, now and in future they will continue to arise in just such a stream of formation and destruction without limit. On this basis, Buddhist scholars assert as a general principle that one cannot posit that the formation and destruction of world systems began at a specific time nor will they end at some appointed moment. 
Excerpted from The Physical World by His Holiness the Dalai Lama