I also thought I saw a documentary or read something about rudimentary transfusions carried out in battlefield hospitals around the time of the Crimean War (??). It still would've been pretty dangerous around Alexei's time regardless--I don't know when they started learning about blood type compatability , contagious diseases and Rh compatability.
OK, I did some research:
Catholic authors take pains to discredit the story of Innocent VIII's deathbed. Supposedly as the Pope sank into a coma, the harrowing story was told that, at the suggestion of a Jewish physician, the blood of 3 boys was infused into the dying pontiff's veins. They were ten years old, and had been promised a ducat each. All three died. Historians of medicine note this event as the first reported historical attempt at a blood transfusion. [I find this HIGHLY doubtful and probably with some anti-Semitic overtones] With Harvey's discovery of the circulation of the blood, more sophisticated research into blood transfusion began in the 17th century, with successful experiments of transfusions between animals. However, successive attempts on humans continued to bring death. The first fully-documented human blood transfusion was administered by Dr. Jean Baptiste on June 15, 1667. He transfused the blood of a sheep to a 15-year old boy (the boy later died, and Baptiste was accused of murder). Only in the first decade of the 19th century was the reason for such death found in the existence of blood types, and the practice of mixing some blood from the donor and the receiver before the transfusion allowed a greater number of successes. While the first transfusions had to be made directly from donor to receiver before coagulation, in the 1910s it was discovered that by adding anticoagulants and refrigerating the blood it was possible to store it for some days, thus opening the way for blood banks.