Definition - Pushkaravarta — (from Pushkara, i.e. water, and vrita, to have place in, i.e., a watery cloud) a name for the flood clouds of the world-destroying deluge.
Vayu Purana Chapter 51 Verse -37 - Tells us that they are class of clouds said to cause death or famine. And are generated from peak of the mountains.
एते योजनमात्राच्च सार्धार्धान्निष्क्रुतादपि |
तेषां धारासारा: प्रकिर्तिता :||
पुष्करावर्तका नाम ये मेघा: पक्षसंभवा: || Vayu 51.37||
These clouds shower forth from a distance of a Yojana , half a yojana
or one forth of yojana . They rain in continuous torrents
Pushkaravartakas (an epithet of a class of clouds said to cause death of famine ) are the clouds that get generated from the peaks
Pushkaravarta Clouds are also Mentioned in Brahmnada Purana
Book 1- Part 2-Chapter 22- Verse 40. - Chapter Name - The Divine Luminaries , Varities Of Clouds . Divisions of Stratosphere. The Chariot of Sun God. - Page no. 218
Brahmanda Purana gives us the detail explanation about them , giving us the defination defination of Pushkaravarta clouds and tells us that "The clouds called Pushkaravarta are born of the wings of mountains . They increase thier size competing with the water and that is the reason they are called Pushkaravarta clouds ( Revolving Pushkaras) i.e. revolving clouds with water.
Here are the excerpts from Brahmanda Purana chapter.
40 Those clouds named Pushkaravartakas are born of the wings. Its conception is threefold .The shower is glorified as continuous
42 Those clouds named Pushkara (were born of those wings ) . They increased in size competing with waters . For the reason , they are
termed Pushkaravartakas (Revolving Pushkaras) .
43 They Assue many forms. Their thundering sound is very terrible .They are the creation of torrential rain at the end of the Kalpas.
They control the Sarhvartaka fire (which burns down the universe then.
44 These (clouds) shower rain at the close of the Yugas ( Thus) those third (set of clouds) have been recounted . They have many
forms, configurations and aggregates. They fill the earth.
Here you will find original sanskrit shlokas and below is the screenshot.
Conclusion - The puranas tells us that the Pushkaravartakas are the clouds shower rain at the close of the Yugas . They are the creation of torrential rain at the end of the Kalpas. And basically they are the flood clouds of the world-destroying deluge.
Note- Your basic enquiry is about the story. Here I have provided basic answer about the mentioning of clouds from puranas. And haven't found any story yet ,but will update the answer soon. So consider this is not the exact answer but a part of that.