Is there any scripture which mentions these smaller cycles of Kaala?
In Vana Parva (Markandeya-Samasya Upa-parva) of The Mahabharata, Sage Markandeya explains the Yuga cycle to Yudhishthira as follows:
"Markandeya replied, 'Indeed, I shall explain all, after having bowed down to that Self-existent, Primordial Being, who is eternal and undeteriorating and inconceivable, and who is at once vested with and divested of attributes. O tiger among men, this Janardana attired in yellow robes is the grand Mover and Creator of all, the Soul and Framer of all things, and the lord of all! He is also called the Great, the Incomprehensible, the Wonderful and the Immaculate. He is without beginning and without end, pervades all the world, is Unchangeable and Undeteriorating. He is the Creator of all, but is himself uncreate and is the Cause of all power. His knowledge is greater than that of all the gods together. O best of kings and pre-eminent of men, after the dissolution of the universe, all this wonderful creation again comes into life.
Four thousand years have been said to constitute the Krita Yuga. Its dawn also, as well as its eve, hath been said to comprise four hundred years.
The Treta-Yuga is said to comprise three thousand years, and its dawn, as well as its eve, is said to comprise three hundred years.
The Yuga that comes next is called Dwapara, and it hath been computed to consist of two thousand years. Its dawn, as well as its eve, is said to comprise two hundred years.
The next Yuga, called Kali, is said to comprise one thousand years and its dawn, as well as eve, is said to comprise one hundred years. Know, O king, that the duration of the dawn is the same as that of the eve of a Yuga.
And after the Kali Yuga is over, the Krita Yuga comes again. A cycle of the Yugas thus comprised a period of twelve thousand years.
A full thousand of such cycles would constitute a day of Brahma.
Note that he doesn't use any 360x multiplier nor calls those years deity-years, so we can assume them to be normal human years.
Which scripture mentions the longer cycles of Kaala?
Interestingly, in Shanti Parva (Mokshadharma Upa-parva), those same years are referred to as "years of the deities". Bhishma here recollects the conversation, Sage Vyasa had with his son Suka, to explain to Yudhishthira the concept of "divisions of time":
"Vyasa said, 'Only Brahma, which is without beginning and without end, unborn, blazing with effulgence, above decay, immutable, indestructible, inconceivable, and transcending knowledge, exists before the Creation.
The Rishis, measuring time, have named particular portions by particular names. Five and ten winks of the eye make what is called a Kashtha. Thirty Kashthas would make what is called a Kala. Thirty Kalas, with the tenth part of a Kala added, make what is known as a Muhurta. Thirty Muhurtas make up one day and night. Thirty days and nights are called a month, and twelve months are called a year. Persons conversant with mathematical science say that a year is made up of two ayanas (dependent on sun's motion), viz., the northern and the southern. The sun makes the day and the night for the world of man. The night is for the sleep of all living creatures, and the day is for the doing of action. A month of human beings is equal to a day and night of the Pitris. That division (as regards the Pitris) consists in this: the lighted fortnight (of men) is their day which is for the doing of acts; and the dark fortnight is their night for sleep.
A year (of human beings) is equal to a day and night of the gods. The division (as regards the gods) consists in this: the half year for which the sun travels from the vernal to the autumnal equinox is the day of the deities, and the half year for which the sun travels from the latter to the former is their night. Computing by the days and nights of human beings about which I have told thee, I shall speak of the day and night of Brahman and his years also.
I shall, in their order, tell thee the number of years, that are (thus) for different purposes computed differently in respect of the Krita, the Treta, the Dwapara, and the Kali yugas. Four thousand years (of the deities) is the duration of the first or Krita age. The morning of that epoch consists of four hundred years and its evening is of four hundred years. (The total duration, therefore, of the Krita yuga is four thousand and eight hundred years of the deities). As regards the other yugas, the duration of each gradually decreases by a quarter in respect of both the substantive period with the conjoining portion and the conjoining portion itself. (Thus the duration of the Treta is three thousand years and its morning extends for three hundred years and its evening for three hundred). The duration of the Dwapara also is two thousand years, and its morning extends for two hundred years and its evening also for two hundred. The duration of the Kali yuga is one thousand years, and its morning extends for one hundred years, and its evening for one hundred.
[Footnote] 156:1 The Krita extends in all for 4,800 years. The Treta for 3,600; the Dwapara for 2,400; and the Kali for 1,200. These are, however, the years of the deities. Verses 15-17 and 20-21 occur in Manusmriti, Chapter I.
So as per the above explanation, each Kali-yuga is 1200 deity-years (or divine years) = 432,000 human years. Vishnu Purana also agrees with this explanation.
How many kinds of Yugas are there? by Satya Sarada Kandula.