Apologies for the broadness of this question, I do not mean to imply that the artistic traditions of two massive cultures over centuries are equivalent or interchangeable! I just know so little about this topic I’d be grateful for answers on either subject.
In European art, particularly in the early modern era, the pastoral is a genre in painting, poetry, drama, music, and more. It depicts an idealized look at agricultural, rural life, often through stock characters and scenes (your shepherds and shepherdesses discoursing or falling in and out of love, your satyrs and so on). It’s often nostalgic, casting a longing eye on what the (often urban) artist or audience perceives as a more wholesome, simple life. It’s all over the place in European art.
Does such a tradition exist in China or India, societies with their own classes of urban artists and professionals, societies that, like Europe, were largely agricultural until quite recently? How did they perceive and depict the pastoral life in the premodern era? Were artists there similarly nostalgic, crafting idealized depictions of a simpler life?
There is a major genre of East Asian art literally referred to as "rivers and lakes." These works, like the European pastoral, extol the beauty and simplicity of nature explicitly opposed to the conflicts and corruption of the actual world they live in.
But of course, there are major differences between the East Asian look at the simple world of nature and the pastoral. I know most about the Early Modern Korean poetic tradition, so I'll draw on that to outline some themes in the East Asian tradition. In particular, a major differences between the pastoral and the "rivers and lakes": East Asians generally opposed rural, natural simplicity not with urban life specifically, but with service in the government and success more broadly.
In Early Modern Korea, great poets belonged to the yangban and (more rarely) chungin classes. Both classes derived their legitimacy from government service. The very word for the Korean nobility, yangban, means "two (yang) halves (ban)" -- that is, the two divisions (military and civilian) of the Korean government. It was the fervent desire of each and every nobleman to pass at least the lowest degree of civil service examinations and attain the position of licentiate. The chungin, a sort of middle class, were also often tied to government service either as clerks in local governments or as professionals recognized by the state. Furthermore, the Confucian tradition adhered to by the Korean elite extolled ch'ung, loyalty to the state and monarch, as a fundamental virtue.
This means that the most basic conflict in many "rivers and lakes" poems was not between a rural and urban lifestyle, but between the poet's desire for the simple life of nature and the countryside and his obligation to serve the king and guide the realm to a better path. This is best illustrated by the first four stanzas of sixteenth-century poet Kwŏn Homun's Eighteen Songs of Leisure. Here's a loose translation:
I sought but loyalty and piety;
Without these things we are as birds and beasts;
With this in heart I wandered for ten years.
--
Such being my plans, fame came much late to me;
My carts of books once anxious for success
Are vain before the flowing stream of time.
--
Be then they vain; the woods and pools are good,
You wuxin fowl and fish in leisure go,
Let me forget the world and follow you.
--
For rivers, lakes, abandoned is my lord;
And for my lord, abandoned are my joys;
A crossroads where I must now choose my trail.
Kwŏn opposes two things here. On one side, we have "fame," "carts of books" (to study for the examinations), and "my lord." On the other, we have the Confucian virtues "loyalty and filial piety," the pursuit of which the second stanza implies precludes fame, "the woods and pools" and their occupants, wuxin, and "my joys." Wuxin here means the Buddhist concept of "no-mind" (wu "no," xin "mind"). In other words, the birds and fish have transcended the petty greed, desire, and hatred that plague the human mind and have reached a state of sagely wisdom.
This is the central theme of many "rivers and lakes" works; success in the "real" world is inimical to the pursuit of a virtuous life, which can be best attained by immersing oneself in nature, but the poet also has the irremovable burden of having to pass the examinations and enter government service to care for the commonwealth of his realm and the welfare of his king.
This theme has many variants. One striking example comes from the putatively fifteenth-century poem Song of Four Seasons in Rivers and Lakes, which both extols a simple life surrounded by nature and exalts the king whose benevolent reign has allowed the poet to live such a leisurely life. Indeed, every stanza ends with the refrain "such X, too, a blessing from my king" (이 몸이 X해옴도 亦君恩이샷다):
When spring is on the streams, a frenzied joy,
Sit on the riverbanks with fish, with wine,
Such leisure, too, a blessing from my king.
--
When summer's on the streams, a quiet house,
From my dear friend, the river's waves, a breeze,
Such coolness, too, a blessing from my king.
And so on. Other poets, meanwhile, sought a more definitive break with the "ruddy dust" of government and its factionalism. Consider the final stanza of Yi Hyŏnbo's Song of the Fishing man:
I see Chang'an a thousand leagues away,
That on my barge I never could forget;
Enough! 'tis not my work to save the world.
The political mission of the Confucian scholar-official was often said to be that of "saving" or "managing the world." In the end, Yi chooses his simple pleasures in nature over his social class's political duty toward the world.
Other poets described nature not as opposed as government service specifically, but to fame, conflict, and poverty generally. This was especially the case with eighteenth- and nineteenth-century professional poets, who were much less likely to have personal experience in a high government post. Here's a typical example, the final stanza of the probably eighteenth-century Song to Spring (claimed to be from the fifteenth century, though literary analysis suggests this is unlikely):
But fame would shun me, fortune shun me too,
Blue wind, bright moon; what other friend have I?
These roads and dusty bowls leave no pretense,
Be it! A hundred years of joy is mine.
The poet opposes fame and wealth generally, not government service per se, to the true happiness of living a simple life. A more explicit opposition between simple nature and corrupting wealth is made more explicitly in the following poem:
I went to sell away my poverty,
But every bargain needs something for free:
"Give me your hills, your moon." "I never shall."
Of course, this wasn't something fundamentally different from the more government-opposed River and Lakes poems. In Early Modern Korea, wealth and especially status were closely tied to government recognition, insofar as even affluent merchants bought yangban genealogies or sought to pass the lowest degree of civil service examinations.
In sum, Rivers and Lakes poetry extolled rural nature in opposition to success and power in the real world, whether political, social, or economic. There was little sense of an opposition between urban and rural landscapes, or the idea that the rural world was yielding ground.
So far, we've discussed poems where the focus is on nature, not on rural people. This would seem like another big difference with the pastoral. But as it turns out, there are plenty of works that extol the simplicity of rural life. The tone isn't necessarily nostalgic, but it is often critical of the elite that's unaware or uncaring of the lives of simple rural folk. The best example is the 1664 Eight Songs of Country Households, by Yi Hwi'il. I quote the first five stanzas:
Though spurned by the world, by the furrows I age,
With no news from afar; I know not what I do,
Yet my love for the realm still prays for the crops.
--
A farmer comes, tells me that spring has arrived,
Takes this family's plow and that family's plow,
"Enough! Not me first! To help others is best."
--
In the heat of the summer the fields are on fire,
As we weed the furrows, our sweat wets the ground.
Alas! Who remembers the work in each grain?
--
In autumn, see the crops, good are they, very good,
Our own strength's own work is of far better taste,
Why envy those men with their steeds and their grain?
--
At night, weave the mats, at day, cut the grass,
And fix our thatched houses and tend to the tools,
Next year, when spring comes, put our minds to our work.
Though Yi describes a rural community, and not the natural landscape that we saw above, his themes aren't radically different. He emphasizes that the simple life of these poor villagers, marked by solidarity, neighborly goodwill, and diligence, is superior to that of the rich and successful. He addresses the contradiction between his current status as a petty countryside gentleman (Yi didn't actually work in any capacity, though he did live with the villagers in the poem and wrote this piece for the village children) with his social obligation to participate in governance by mentioning that he has been "spurned" by the world and by wishing for a year of agricultural prosperity, not for the peasants' sake but for the nation's. If there is a difference, it's that the focus is wholly on this rural community rather than nature.
Other poets depicted an even more romantic and idealistic view of rural life, emphasizing both the diligence and leisure of villagers in the midst of pristine nature. Eighteenth-century poet Wi Paekgyu's Farming Song is a typical example of this romantic portrayal of farmers:
[Stanza 1]
The sun shines off the west; the clouds are low,
The grass's green's but deepened after rain,
Enough! We have our work; weed, weed them all.
--
[Stanza 4]
All the sweat that can fall, falls; all the sun that can shine, shines.
Chests open to the wind, our whistles sound,
And there, that man -- he seems to know a thing.
--
[Stanza 8]
The old are drunk, the children all in mirth,
And as the wine cup goes once more around,
Who now delays our dance with drum and song?
But again, it's difficult to speak of a nostalgia for a disappearing thing per se. But it's worth remembering that Early Modern Korea was not a very urbanized society; while 10% of the 18th-century European population lived in cities with more than 50,000 people, less than 2% of Koreans lived in Seoul, the single city with more than 50,000 people. For Koreans, rural life must have seemed eternal.