Lithium-ion batteries
Yes, lithium ion batteries works with the principle of intercalation and deintercalation of ions from cathode to anode and anode to cathode respectively. To be specific for intercalation, lithium (in cathode) having high oxidation potential oxidize, loosing an electron which goes to the current collector to anode. So lithium ion get positive charge. Since the battery source you use also has positive terminal hence repulsion between ion and positive terminal occurs. Thus positive ion of lithium goes towards anode, which has layered structure thus result in intercalation. Another way of seeing it is because of difference in the concentration the diffusion of ions occur or you can also consider the fact charge neutrality.