The phrase "a tiny amount of their" is correct and usable in written English.
It can be used when referring to a small quantity belonging to a group or individuals.
Example: "The researchers found that a tiny amount of their funding was allocated to community outreach programs."
Alternatives: "a small portion of their" or "a minimal quantity of their".
I have no problem with those found guilty of a serious criminal offence being on the database, especially those in prison - it seems small beer that the state holds a tiny amount of their DNA on file when the primary clump of their genes is being held at Her Majesty's pleasure.
However, mitochondria have a tiny amount of their own DNA so any resulting child would have genetic information from three people.
When mosquitoes bite, they inject a tiny amount of their saliva (less than a microliter), which contains a specialized, potent cocktail of molecules that numb the pain of the bite and stop the blood from clotting.
They spoon a tiny amount of their drugs into a plastic baggy, then toddle off to the bar for 20 minutes.
It is distressing, to say the least -- during fashion weeks I've seen women not eat at all, I've seen women becoming riddled with anxieties about consuming a tiny amount of sugar because their skin will break out from all the stress.
These nanoparticles absorbing a tiny amount of n-octanol in their porous silica over-layer are homogeneously dispersed into a bulk aqueous phase (pH 7.40) containing an organic compound prior to magnetic separation.
And now, marketers can pay to do market researches on Voice Polls, and users will get a tiny amount of money to answer their questions — the startup probably takes a cut.
To save money, some manufacturers put a a tiny amount of an ingredient in their formula just so they can have it on the label, even though the actual dose may be too small to actually have any significant effect on you.
Small transistors that control the flow of electrical current to the material's crystalline surface are used to apply a tiny amount of heat to individual crystals, modifying their state and thus their electrical resistance.
Indonesia's political parties get a tiny amount of state funding, so have to generate their own cash through membership fees (bugger all) and selling promises to opportunistic private sector players (much more profitable).
