We study the dynamics of polymers and elastic manifolds in nonpotentia
l static random flows. Barriers are found to be generated from combine
d effects of elasticity, disorder, and thermal fluctuations. This lead
s to glassy trapping even in pure barrier-free divergenceless flows, w
ith anomalously small response to an applied force. We find a new reno
rmalization-group fixed point at finite temperature and compute the ro
ughness, dynamical, and response exponents for directed and isotropic
manifolds.