For this year's Janmāṣṭamī blog I'd like to quote some arguments the Goswāmīs had to claim that Kṛṣṇa actually appeared in Gokula (as well as in Mathura). From Śrīla Rūpa Goswāmī's Laghu Bhāgavatāmṛta (5.442-450, 454, 461):
442 Līlā Puruṣottama Śrī Kṛṣṇa, whose vilāsa-form is the great Nārāyaṇa, desired to appear in Gokul and Mathurā, so He first made Saṅkarṣaṇa appear. Eventually He would let the other members of the caturvyūha, Pradyumna and Aniruddha, appear as well, but first He Himself appeared in the heart of Vasudeva Mahāśaya.
443 To relieve the burden of the earth and petitioned to appear by the devatās, He appeared in the 28th catur-yuga, at the end of that particular Dwāpara Yuga. In his commentary Śrī Baladeva Vidyābhūṣaṇa adds a verse from the Matsya Purāṇa which says that Lord Vasudeva will appear in 3 forms at the end of Dwāpara Yuga - as Vyāsa, Rohiṇī's son (Balarāma) and Keśava (Kṛṣṇa). Aniruddha, who reclines in the milk ocean, appears in the heart of Vasudeva in Mathurā, merging with Swayam Bhagavān Śrī Kṛṣṇa there. From Vasudeva's heart Śrī Kṛṣṇa penetrates Devakī's heart.
444 Nourished by the nectar of Devakī-devi's vātsalya prema, Hari grew within her heart like the waxing moon.
445 Thus, in the middle of the night on the eighth day of the dark lunar quarter in the month of Bhādra (August-September), through the agency of Yogamāyā, Śrī Kṛṣṇa appeared from Devakī-devi's heart into the lying-in chamber in the Mathurā jail.
446 Under Yogamāyā's spell Devakī and all others present thought that Kṛṣṇa had taken birth happily in the ordinary human way.
447 Śrī Baladeva Vidyābhūṣaṇa quotes from the Viṣṇu Purāṇa that Devakī beheld Kṛṣṇa in a two-armed form but the Bhāgavata (10.3.39) clearly says catur-bhujaṁ śaṅkha-gadādyudāyudham. How is that contradiction resolved? Both in the four-armed and two-armed form Kṛṣṇa acts like a human being - though He may seem bewildered He is actually omniscient. Even if Kṛṣṇa sometimes reveals a four-armed form the scriptures say the two-armed form is predominant - Rūpa Goswāmī says He never really gives up the feelings, attributes and form of the two-armed form. Why then do the śāstras sometimes make it appear as if the two-armed form is secondary?
448 The answer to that is that when Kṛṣṇa's great prowess is hidden within His two-armed form it may be described as secondary. Nārada told the mourning Yudhiṣṭhir Mahārāja in the Seventh Canto (7.10.48 and 7.15.75) that the Supreme Brahman had lived in his house in a hidden way, in a human form.
449 After Kṛṣṇa's birth Vasudeva brought Him to Gokula, placed Him in the delivery room in Yaśodā's house, took Yaśodā's daughter with him and went out again.
450 Though Kṛṣṇa is Yaśodā's eternal son since beginningless time, in His manifest earthly pastimes He does appear like that. Baladeva Vidyābhūṣaṇa says in his comment that Yaśodā suffered so much labor pain and was so tired that she was not aware of the gender of her baby - she lost consciousness by the influence of Yogamāyā. Viṣṇu appeared to Devakī devī and Yaśodā at the same time. Śrīpāda Baladeva quotes the Harivaṁśa (2.4.11) here:
garbha-kāle tv asampūrṇe aṣṭame māsi te striyau; devakī ca yaśodā ca suṣūvāte samaṁ tadā
'In the eighth month, when the pregnancy was incomplete, both women, Devakī and Yaśodā, delivered simultaneously."
The eight-armed Devi took birth afterwards. In Śrīmad Bhāgavat 10.4.9 Yogamāyā is called the younger sister of Kṛṣṇa, which would indicate that Kṛṣṇa had already been born before her. Śrī Jiva Goswāmī comments on 10.4.9 - śrī visnoḥ śrī ḍevakī yasodayor manasi yugatpat pravistasyeti tasyās tad anujātvam sādhitam anujeti "The fact that Devī is called Kṛṣṇa's younger sister indicates that He entered Śrī Devakī's and Yaśodā's minds simultaneously." Viśvanāth Cakravartī comments: anujā viṣṇor ityanena kṛṣṇasya yaśodā garbhajatvaṁ sūcayati - "The fact that Devī is called the younger sister of Kṛṣṇa indicates that Kṛṣṇa was born from Yaśodā's womb." Just as Vasudeva was lifting Kṛṣṇa up to bring Him from Mathurā to Gokula the goddess Yogamāyā or Aja was born from Yaśodā in Gokula - hence Śukadeva called her Kṛṣṇa's younger sister. Baladeva then quotes Sage Nārada saying in the Ādi Purāṇa - nanda gopa gṛhe putro yaśodā garbha sambhavaḥ - 'In cowherd Nanda's house a son was born from Yaśodā's womb'. If one asks why there is such a secrecy (in the Bhāgavat) about Kṛṣṇa being born from Yaśodā, the answer is - 'It is the Lord's wish' - "I will appear in the house of both Nanda and Vasudeva. I will however remain only in Nanda's house with one form. If I have two forms, Kaṁsa will know that I have taken birth and will persecute both sets of parents. You (Śukadeva) should recite my story so that the secret is not revealed." This is the intention of the Lord. Accepting the Lord's desire, the author wrote the work accordingly. "
The issue is further discussed in verse 454 - 'Some ancient devotees argue that the first of the Caturvyūha, Vasudeva, took birth in Vasudeva's abode and Līlā Puruṣottama Śrī Kṛṣṇa and His Yogamāyā potency appeared in Nanda's house in Gokula. When Yadubar Vasudeva arrived in Gokula to bring Kṛṣṇa he only saw Yogamāyā in the delivery room, not Kṛṣṇa - at that point Vasudeva Kṛṣṇa merged with Līlā Puruṣottama Kṛṣṇa. Śukadeva did not narrate it directly and clearly, though, due to its confidential nature. Then in verse 461, Śrī Rūpa Goswāmī quotes the famous verse vṛndāvanaṁ parityajya pādam ekam na gacchati, which he learned from Śrīman Mahāprabhu, to make the point that if Kṛṣṇa never leaves Vraja He also does not come into Vraja, and therefore he must have been born in Gokula (too).
The cutting of the umbilical cord (nābhi-cheda)-pastime usually shows where which śāstra is at. Brahma Vaivarta Purāṇa canto 4, chapter 9, verse 60, describes this scene after Vasudeva had brought Kṛṣṇa to Gokula:
dhātri tam snāpayamāsa sīta-toyena bālakam
ciccheda nāḍim bālasya harṣād gopyo jayaṁ daduḥ
"The midwife bathed the infant boy with cool water and cut the umbilical cord as the gopīs glorified Him."
It is a bit strange though that this text was never quoted by the Goswāmīs, who all agreed on the fact that Kṛṣṇa was born in Gokula, and must have been eager to prove it from śāstra. This verse would have been a golden opportunity to prove this. Thus the text could be an interpolation. In the Gopāl campu (4.29-30) it is described that there was no more umbilical cord on Kṛṣṇa when the midwife was to cut it in Gokula, which would indicate that it had been cut by Devakī devī in Mathurā already.