This paper reports a novel microfluidic-chip based platform using "phase-transfer magnetophoresis" enabling continuous biomolecule processing. As an example we demonstrate for the first time continuous DNA extraction from cell lysate on a microfluidic chip. After mixing bacterial Escherichia coli culture with superparamagnetic bead suspension, lysis and binding buffers, DNA is released from cells and captured by the beads. These DNA carrying beads are continuously transported across the interfaces between co-flowing laminar streams of sample mixture, washing and elution buffer. Bead actuation is achieved by applying a time-varying magnetic field generated by a rotating permanent magnet. Flagella-like chains of magnetic beads are formed and transported along the microfluidic channels by an interplay of fluid drag and periodic magnetic entrapment. The turnover time for DNA extraction was approximately 2 minutes with a sample flow rate of 0.75 µl s(-1) and an eluate flow rate of 0.35 µl s(-1). DNA recovery was 147% (on average) compared to bead based batch-wise extraction in reference tubes within a dilution series experiment over 7 orders of magnitude. The novel platform is suggested for automation of various magnetic bead based applications that require continuous sample processing, e.g. continuous DNA extraction for flow-through PCR, capture and analysis of cells and continuous immunoassays. Potential applications are seen in the field of biological safety monitoring, bioprocess control, environmental monitoring, or epidemiological studies such as monitoring the load of antibiotic resistant bacteria in waste water from hospitals.