I know that an extracted living heart is preserved for transplantation with a nutrient-rich solution, but I could not answer a query related to maintaining appropriate oxygen supply to the same heart during transportation.
Using a standard preservation method there is no oxygen supply since the heart is arrested and submerged in the preservation solution during transport.
As mentioned in previous comments, common to all preservative solutions is the use of hypothermic cardioplegia. Hypothermia fundamentally reduce intracellular metabolism, decrease rate of enzymatic degradation and thus preserve intracellular energy. Following cardioplegic arrest and surgical exicision of the donor heart, the heart is placed on ice. This technique is referred to as static storage. With an optimal temperature between 4 to 6 degree, static storage provides reliable preservation for up to approximately 6 hours. However, there are reports of satisfactory results in hearts preserved up to 14 hours using this technique [Ref: Obadia JF et al JHLT 1997; 16: 256-9}
Actually the heart is not supplied in oxygen during transportation; myocardial protection is achieved by delivery of cardioplegia at the time of excision (i.e., Celsior solution) and by organ hypothermia (at about 4°C). This allows a 'safe' preservation period of at least 4 hours, up to 6 hours (including the time needed fo implantation in the receiver), although longer successfull preservation times have been reported.
we use of hypothermic cardioplegia, decompresssion of the heart is very important during cardioplegia delivery. transportation in a cold saline at about 4°C without any additive is mostly enough for 4-6 hours.
Use of Cold crystalloid cardioplegia, making sure heart is not distended by draining it well after opening IVC & pulmonary veins/ or LA appendage if lungs are harvested , making sure cardioplegia delivery is good to achieve good diastolic arrest with completely flaccid heart on palpation ( this is very important)and then using ice slush or cold saline irrigation to cool epicardial surface of the heart evenly, packaging explanted heart for static cold storage transportation. A good cold cardioplegic diastolic arrest and proper cold quick transport will keep the heart in good condition for transplantation.
As others have mentioned usually such well harvested & cold preserved heart can withstand 4 - 6 hrs of cold ischemia time.