Recently, continual learning has received particular attention in machinery remaining useful life (RUL) prediction, which enables prognostics networks to gradually improve performance without laborious retraining. Existing studies, however, have the following limitations: 1) The impacts of variable operating conditions are not explicitly considered during continual learning. 2) The continual learnability of prognostics network is limited by the lack of knowledge compression and de-redundancy. To overcome the abovementioned limitations, a novel continual learning framework is proposed for RUL prediction of machinery under variable operating conditions. First, a multi-kernel swarm convolution block is devised to automatically capture degradation features during continual learning without increasing computational consumption, which combines convolutional operations at various scales and focuses attention on appropriate scales based on operating condition information. Then, core space gradient projection is proposed for continual learning of prognostics networks, which mitigates forgetting by guiding gradient descent along the orthogonal direction of the previously input subspace. This approach also ensures the network learnability during continual learning by knowledge compression and de-redundancy to identify core space. The proposed framework is verified using accelerated degradation datasets of rolling element bearings with variable operating conditions. Experimental results show that the proposed framework is superior to some existing continual learning-based prognostics approaches for RUL prediction under variable operating conditions.