Do the mleccha civilizations become stronger than the vedic civilization in kali yuga?
Yes, kali yuga is when the destruction of Vedic civilization and the rise of the Mleccha civilizations happens.
From the Vishnu Purana on the symptoms of Kali Yuga:
The people of the various countries intermingling with them will follow their example, and the barbarians [mlecchas] being powerful in the patronage of the princes, whilst purer tribes [Vedic civilization] are neglected, the people will perish.
Translator H.H. Wilson says in the footnote to that verse:
The commentator, having no doubt the existing state of things in view, interprets the passage somewhat differently: the original is, ###. The comment explains 'strong', and adds, the Mlechchhas will be in the centre, and the Áryyas at the end:' meaning, if any thing, that the unbelievers are in the heart of the country, and the Hindus on the borders: ... All that the text intends, is to represent infidels and foreigners high in power, and the Brahmans depressed.
The Vayu Purana also says that Hindus will adopt Mleccha beliefs in kali yuga, and even inter-marry with them:
'Intermixed with them, the nations, adopting every where barbaric institutions, exist in a state of disorder, and the subjects shall be destroyed.'
The symptoms of Kali yuga are especially found in the Western (Mleccha) civilizations:
Wealth and piety will decrease day by day, until the world will be wholly depraved. Then property alone will confer rank; wealth will be the only source of devotion; passion will be the sole bond of union between the sexes; falsehood will be the only means of success in litigation; and women will be objects merely of sensual gratification ... mutual assent will be marriage; fine clothes will be dignity... Gold, jewels, diamonds, clothes, will all have perished, and then hair will be the only ornament with which women can decorate themselves. Wives will desert their husbands, when they lose their property; and they only who are wealthy will be considered by women as their lords... The minds of men will be wholly occupied in acquiring wealth; and wealth will be spent solely on selfish gratifications. Women will follow their inclinations, and be ever fond of pleasure. Men will fix their desires upon riches, even though dishonestly acquired... The women will be fickle, short of stature, gluttonous: they will have many children, and little means: scratching their heads with both hands, they will pay no attention to the commands of their husbands or parents: they will be selfish, abject, and slatternly: they will be scolds and liars: they will be indecent and immoral in their conduct, and will ever attach themselves to dissolute men.
the Brahmanical thread will constitute a Brahman; external types (as the staff and red garb) will be the only distinctions of the several orders of life;... Amidst all castes he who is the strongest will reign over a principality thus vitiated by many faults... Men of all degrees will conceit themselves to be equal with Brahmans.
So, the conclusion is that kali yuga is the time when mleccha races prosper.