While it would be much better to wait just for answers in Song-Yuan-Ming period China specialists, what I'll post is just a very brief summary for the state of reseach made from just a few articles originally meant for my personal understanding, no specialist in eastern Eurasian history.

 

Some Basic Premises:

1. This topic has been hotly debated among the historians especially since 2010/11 when DNA analysis suggested a possible Chinese provenance of Y. pestis, but no agreement so far: Compare [Hymes 2014] and [Benedictow 2013] or [Buell 2012] that both were very negative to this hypothesis, though on different grounds.
2. Why no conclusion? (1) : AFAIK no contemporary Mongol account like The Secret History of the Mongols, or Ilkhanate texts like Rashid-al-Din Hamadani (d. 1318)'s Compendium of Chronicles (Jāmiʿ al-Tawārīkh) is useful for this topic, so instead we can solely rely on Chinese written sources (see below)
3. Why no conclusion? (2): While several epidemic outbreaks were recorded in the primary sources like official histories of the Chinese dynasties in the 13th and 14th century, almost no direct (detailed) mention (see below) of the bubonic plague or any other disease outbreaks: In most cases, they just told us the outbreaks of the disease (疫) only.
4. Why no conclusion? (3): It would be nearly impossible to explore this problem further with help of DNA analysis due to the burial custom, as I noted in this question thread briefly.

 

The article and monograph of a Chinese historian, Cao Shuji (曹树基) (unfortunately in Chinese), seem in fact to the most up-to-date and comprehensive work of the epidemic outbreaks in China from 13th century to 20th century (Cf. Hymes 2014), in fact far more comprehensive than the chinese work that [Mcneill 1976] and [Abu-Lughod 1989] draw their data. They are also a controversial works, and few historians have taken the idenfication of Cao of the plague to several records at face value (Cf. Buell 2012; Ijima 2009: 11f.), though.

 

While Hymes himself does not accept most of Cao's identification without reservation, he pays special attention to the two entries of outbreaks among Cao's list during the Mongol conquest of China in the 13th century (Hymes agrees to the hypotheses of McNeill and Cao that suppose the Gansu/ Mongolia origin of the plague), and identify them as the likely outbreaks of the plague:

1. The outbreak occurred in Daliang (modern Kaifeng, Henan), the capital of Jin Dynasty (1115–1234) in Northern China after the siege of the Mongols in spring 1232, recorded in official history as well as other accounts (Hymes 2014: 289-91), like Jin Dynasty Doctor Li Gao's account, as following:

'In all it had been half a month since the enemy’s arrival, and after the siege was lifted, not one or two out of ten thousand of the people of the capital did not become sick, and the sick who died followed another without end. At each of the capital’s twelve gates, [the dead] sent out each day were two thousand at most and no less than one thousand at fewest, and this was so for almost three months......It was not only in Daliang that things were this way. Further back, during the Zhenyou and Xingding eras [1213–22], [cities] like Dongping, like Taiyuan, or like Fengxiang were all the same in the illness and death they suffered after their sieges were lifted' (Li Gao, Neiwaishang bianhuo lun, bian yinzeng, yangzheng (1247), in Li Gao 1993: 8–9, taken from Hymes 289f.)

1. The outbreak of 1273 among the Song dynasty supporters against the Mongols, narrated as a prophetic dream of the former chancellor, Chén Yízhōng in official Song Histories, compiled in the 1340s (Hymes 2014: 293f.), as following:

'Before this, Chen dreamed someone told him: "This year a Heaven-sent disaster will spread, and nearly half the people will die; those who take rhubarb will live."......' (Tuotuo et al., Song Shi, translation is taken from Hymes 2014: 293).

 

Main argument of Hymes from these two accounts are twohold:

• Both accounts record the very high mortality ('nearly half'), which had been rarely known in preceding dynasties in China.
• Especially Li Gao's account of 'after their sieges were lifted' strongly suggests that the citizens of Daliang as well as those of other cities got their disease from the Mongols, and his supplementary notes of the possible cause of the disease (I omitted due to its complexity for non-medical specialists) also suggests this epidemic was new to the 13th century Chineses.

Thus, something new, very deadly epidemic like the plague came to China in the first half of the 13th century, together with the invading army of the Mongols, he argues.

 

That is almost all we have in our hands for the 13th century eastern Eurasia and I hope these accounts are the closest OP is looking for.

 

References:

• Abu-Lughod, Janet L. Before European Hegemony: The World System A. D. 1250-1350. Oxford: OUP, 1989.
• Benedictow, Ole J. 'Yersinia pestis, the Bacterium of Plague, Arose in East Asia. Did it Spread Westwards via the Silk Roads, the Chinese Maritime Expeditions of Zheng He or over the Vast Eurasian Populations of Sylvatic (Wild) Rodents?' Journal of Asian History 47-1 (2013): 1-31.
• Buell, Paul D. 'Qubilai and the Rats'. Sudhoffs Archiv 96-2 (2012): 127-44. http://www.jstor.org/stable/43694442
• Hymes, Robert. 'Epilogue: A Hypothesis on the East Asian Beginnings of the Yersinia pestis Polytomy'. In: Pandemic Disease in the Medieval World: Rethinking the Black Death, ed. Monica Green, pp. 285-308. Kalamazoo, MI: Arc Medieval P, 2014.
• McNeill, William. Plagues and Peoples. New York: Anchor, 1976.
• IJIMA, Wataru. Chinese History seen from Epidemics: Public Health and Eastern Asia. Tokyo: Chuo-Koron Sha, 2009. (in Japanese)

[edited]: fixes typo.