So I can only really talk about this from the perspective of England (and somewhat Scandinavia). Fortunately its an extremely useful case study with a wide variety of cemeteries and burial practices all very well documented and covering a wide area both temporally and chronologically, as well as in belief systems.

Now before moving on it is important that we understand one thing. Christians utilized grave goods in their burial rituals and pagans also inhumed/cremated bodies with few goods accompanying the body. It is rarer, all things considered, for Christian populations to be associated with grave goods accompanying the dead, but it is not unheard of. There is a certain assumption by many people that the burial practices of a population did not change without some sort of great change in belief systems. This is utterly untrue and before moving on we need to understand that there was no immutable truth that pagan burials included goods and Christian ones did not.

In England at the end of the Roman presence there and the arrival of "Anglo-Saxon" migrants, there are a variety of practices observed in burials among the various populations. Inhumation, both furnished with extensive grave goods, and simpler, as well as cremations, again with and without grave goods, are all seen across the country side. Efforts to establish one practice or another as the exclusive domain of one population over another are fraught and really in the end are untenable. There was no rigid dividing line as might be imagined. The pagan Anglo-Saxons did not all cremate or inhume with grave goods, the Christian Britons did not all have unfurnished inhumations in church yards.

The range even between furnished inhumations is quite extreme. On one hand there are any number of burials where the dead person was accompanied by a few small dishes, some bones or teeth/evidence of food offerings, and maybe some jewelry or a weapon of some sort. On the the other hand there is the relative extravagance of the Sutton Hoo burial were weapons, a suit of mail, whole animals, and even a boat were interred. Nor can we comfortably put all of these burials firmly into the realm of "pagan". The Sutton Hoo example famously contained silver jewelry bearing the names of Saul and Paul in addition to the more obviously pagan elements of animal internment.

In England at the end of late Antiquity and the early medieval period the classic "Christian" burial is an unadorned inhumation with the body facing east and nearby a church, but this was not the exclusive domain of Christian burial. Indeed this pattern had its roots earlier in history, before conversion began to really pick up steam in England. Nor did it mark the end of burials with rich adornments. Robin Fleming (whose book Britain after Rome I am drawing heavily from) points to the example of St. Cuthbert whose tomb was not only furnished with jewelry but even opened up later on in order for more to be added.

Christian attitudes towards grave goods were not uniformly negative. The mentioned example of St. Cuthbert is certainly a spectacular example, but there are other, less extravagant graves, that fall firmly within the Christian time frame but still exhibit grave goods such as specially placed rocks, knifes, and small bits of jewelry. Even in Christianized literature such as Beowulf, the grave goods and cremation of Beowulf is remarked on with no commentary, despite the text having clearly undergone some shifts by the Christian author.

So if there is not really any differentiation between belief systems based on funeral goods, where does the practice arise from?

Guy Halsall, in Worlds of Arthur argues that it is a symptom of social stratification in uncertain political times. Burying family members with grave goods was a mark of social standing and wealth far more than it was a marker of religious affiliation. In the period of economic collapse in Gaul and Britain, burying your loved ones, regardless of belief, with jewelry, weapons, etc... was an important social signifier. He also connects this to economic crisis in the Saxon homelands in Germany, the Low Countries, and Denmark.

He takes this a little farther and even argues that the inhumations found in Saxon/Frankish cemeteries was in imitation of Roman custom as a signifier of social importance. Roman connections after all being valuable and worth flaunting to the rest of society.

By the 7th century however grave goods were across the board becoming rarer and rarer, even in communities believed to have still been pagan. This wasn't necessarily due to the conversion of the Anglo-Saxons to Christianity, but was a rather response to changing social and economic factors at the time.

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Sources:

I've used Robin Fleming's Britain after Rome and Guy Halsall's Worlds of Arthur pretty heavily in this response. I briefly mentioned Beowulf as well for its own description of burial practices.

If you're interested in more elaborate descriptions of heathen burials with spectacular grave goods, there are a number of sources on the Norse that I can recommend, and their own process of conversion and its effects on burials is also fascinating. For a primary source there is Ibn Fadlan's account of the Rus.