Does the bq25504, bq25505 or bq25570 provide constant current charging?

Because the bq255XX was designed to be powered from a high impedance input source, the bq255XX can only take energy from the source in short bursts before the source collapses below it's maximum power point.  This means that the charge current provided to the battery/load will be in bursts as well. Thus constant current charging is not practical from a boost converter-based charger that is powered from a high impedance source.