While the heart is not beating.
Sometimes exchange can involve removing and exchanging small amounts of blood at a time until the desired percentage has been achieved with two catheters in separate and usually oppositional (e.g., right and left leg) veins to allow removal and transfusion simultaneously to equalize over a long period of time or using a central venous line to administer the treatment more rapidly. But, if a full exchange is the goal, as in a major sickle cell crisis or where it is critical to remove all infected (or in the case of some toxins, "affected" ) blood to replace with tested blood/plasma, then cardiopulmonary bypass (CPB) may be needed.
With the parasite Babesia in overwhelming infection, it may still be possible to do a partial exchange transfusion (without CPB) that may be less than complete but sufficient to stabilize. The patient would then receive appropriate medications to clear the blood of any remaining parasites--usually atovaquone and azithromycin in humans, doxycycline + enrofloxacin + metronidazole in dogs (and being looked at in humans). Yeah, I've seen a few cases, albeit none diagnosed so late as to require blood exchange.
Probably TMI, but that's your answer. 
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