This paper introduces an online motion rate adaptation scheme for learned video compression, with the aim of achieving content-adaptive coding on individual test sequences to mitigate the domain gap between training and test data. It features a patch-level bit allocation map, termed the $\alpha$-map, to trade off between the bit rates for motion and inter-frame coding in a spatially-adaptive manner. We optimize the $\alpha$-map through an online back-propagation scheme at inference time. Moreover, we incorporate a look-ahead mechanism to consider its impact on future frames. Extensive experimental results confirm that the proposed scheme, when integrated into a conditional learned video codec, is able to adapt motion bit rate effectively, showing much improved rate-distortion performance particularly on test sequences with complicated motion characteristics.