A method comprises receiving content from a source, and generating multiple content streams each comprising the received content. Each content stream includes segments comprising respective portions of the received content. Different segmenting is utilized in generating respective ones of the content streams such that the segments in each of the content streams are offset in time relative to corresponding segments in the other ones of the streams. For example, generating the multiple content streams may comprise segmenting n content streams such that each content stream comprises segments of length Ts with corresponding segments in adjacent ones of the content streams being offset from one another by an offset Tg. Such an arrangement ensures that latency shift upon switching between one of the content streams and another content stream in a different set of multiple content streams at a client device can be limited to a value approximated by Tg.