You can play your Dm 3rd positions licks over 11th position in Bbmaj
This demonstration uses a C diatonic harp. The example is played through the EHX HOG and Boss DM2 and the trailed out with an EV660 mic though a Harp Gear F1 amp.
The backing tracks I made can be found here:
Chord name --------------------- Notes used
Bb major 7th --------------------- Bb D F A
Bb sus7 -------------------------- Bb .. F A
D minor ------------------------------ D F A