Let's say Ancient Egypt, since it's where the subject of grave robbery tends to come up most often. It seems like a lot of graves were broken into and raided of treasures in antiquity. But you're some regular Joe who now has some really valuable (and possibly recognisable) goods on your hands. How do you make money from them without being noticed? Was there a black market? Did they just try and melt down precious metals and remove jewels and so on?
About most periods we know very little or nothing at all. The most interesting period (as far as the looting of high status and royal burials) is the looting of tombs in and around of the valley of the kings. Because here we have quite a lot of written documentation of them in the form of trials papyri and a lot of short to quite long graffiti.
As far as we can tell, many tombs were looted just after the burials weree made. (e.g. Tutanchamun, Thuya and Yuja). Sometimes possibly by the very men who organised the burials. In this fase we see mainy the looting of perishable paterials of high quality: glass wares, some types of offerings, textiles etc. That these ended up in the hands of people already handling funerary goods might have raised little suspicion but we do not know.
We know from papyri recording trials (e.g. pap. Leopold) that looting the royal tombs became quite widespread by the beginning of the 20th dynasty. From these sources, we can see that the looting was already well organised: people specialised in digging tombs were now looting them, and they seem to have been backed by persons rather high up in the Egyptian society, who could sell the looted goods. There was also bribery of for example ferry men to take the to the valley and possibly any guards and witnesses encounterd.
In this stage we see the destruction of the tomb contents, mainly to extract valuable metals. (eg. gilded wooden coffins were burned and the gilding melted into small gold strips or bars)
By the end of the 20th dynasty and the beginning of the 21th dynasty, looting had become state buisiness.
In the tombs we now see short graffity of officials doing "inspections" of their state. At the start this might have been a genuine attempt to see which tombs had been looted.
In the next stage we see that Pharaos and High priests of Amun organised the so called "osirification" of the previous Pharaos.
It used to be the consensus that these Pharaos and High priests were collecting the remains of the Royal burials after heavy plunderings and re-wrapped them and put them into safe , well hidden tombs. A deed of great piety indeed.
Recent research has put another light on these "Osirifications". This was a period of Egpyt in economical distress. The Royal tombs were good sources of income for the impoverished state.
The tombs were systematically plundered at this time, by qualified workmen in service of the state. We see that some of the tombs might have even turned into workshops were tomb equipment was being sorted, repurposed or being strippid of their valuable metals, gems etc.
This was done in a highly systematic way: the heavy gilding of wooden coffins were carefully hacked off. The mummies were re-wrapped in order to get at the jewels in the original wrappings. After that they were put into the wooden coffins, now stripped of their valuables and short official texts written on them containing such details as their names, who organised the osirification, from which tomb they came.....
Funerary equipment that had no or little monetary value in itself was converted for other uses: a good example are royal shabtis, which were converted into small Osiris statues for worshipping by re-inscribing them and mouting them on wooden bases. At the other end of the scale we see for example that the granite sarcophagus of Mernepthah was dragged out of the tomb and shipped to Tanis for re-use by the (local) Pharaos there: this indicates complicity of the state(s) at a large scale, and a trade on a huge scale of useful objects from the tombs.
H. Willems has recently written an article that these mummifcations and the repositioning of old pharaos to more hidden tombs might have even had had a ritual context and was also used as a legitimsation of the current rulers of Thebes. From graffity he could reconstruct that the old royal mummies were moved in the new tombs of the local rulers in official processions, linked to the procesions the the new burials being made. (sadly enough this article is only available in Dutch at this moment: H. WILLEMS, ‘De herbegraving van koninklijke mummies in de 21e dynastie’, Phoenix 62.1 (2016), p. 41-55.
In summary there was first a phase of robbery just after the burials, later there were robberies by small but well organised and connected gangs but all of this was still highly illegal. Later on the plundering became almost industrialised with materials being recuperated for their monetary value (Gold, etc...) but not destroyed at random and the re-use of items with little monetary value for new burials etc... And was now "legalised"