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You should be able to transit without visa according to Timatic:
Summary
No, The travel documents held are not sufficient for the traveler's journey, or they may need additional documents as follows:
Type: Critical Russian Federation - Destination Visa
Visa required.
Additional information:
Visitors are required to hold documents required for their next destination.
Visitors are strongly recommended to hold a health/travel insurance to cover for any medical expenses.
Important:
Visitors not holding return/onward tickets could be refused entry.
Type: Ok Russian Federation - Destination Health
This information is for guide purposes only. Other health organisations may recommend alternative precautions.
Vaccinations not required.
Type: Notice Kazakhstan - Transit Visa
Visa required.
Transiting without a visa is possible for:
Holders of onward tickets for a max. transit time of 24 hours.
Type: Notice Kazakhstan - Transit Health
This information is for guide purposes only. Other health organisations may recommend alternative precautions.
Vaccination against yellow fever required if arriving within 6 days after leaving or transiting countries with risk of yellow fever transmission .
Exempt from Yellow Fever vaccination:
Children under 9 months of age.
Passengers transiting Kazakhstan provided not leaving the transit area.
Passengers transiting countries with risk of yellow fever transmission provided not leaving the transit areas.