The radial motion of unichiral magnetic bubbles in an in-plane field has been investigated using transient photography. It is found that when a 20-Oe in-plane field is applied to a unichiral bubble, the wall section whose center spin is opposed to the in-plane field is initially immobile during a bias-field step pulse. This immobility lasts for about 25 nsec, depending on the pulse field. The normal wall section, whose center spin is along the in-plane field, moves with a drive-dependent velocity on the order of 20 m/sec. The velocity difference between the two walls occurs in films of all types, but is particularly prominent in the film with moderate damping (α=0.14) that is used here. Numerical calculations show that the in-plane field produces a different static-spin structure in each of these two wall sections. In the normal wall section, the in-plane field merely reduces the total twist due to the surface stray field. However, in the wall whose center spin is opposed to the in-plane field, a statically stable 2π horizontal Bloch line is nucleated when the in-plane field is above a threshold, which is 12 Oe for the film used here. The velocity of the normal wall is compared with that predicted by a one-dimensional model, and they are found to be in agreement for pulse fields less than 8 Oe. Numerical calculations of the velocity of the wall with the 2π HBL are made using Hubert's finite-difference method. These calculations confirm that the wall section is essentially immobile until the HBL approaches the film surface. When the pulse field is greater than 8.6 Oe, punch-through of the 2π HBL occurs, leaving two opposite-winding VBL's. An implanted layer suppresses punch-through.