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We prove that if this process satisfies a natural \u201cnon-signalling\u201d condition between past outputs and future side information, the min-entropy of the outputs <jats:inline-formula><jats:alternatives><jats:tex-math>$$A_1, \\dots , A_n$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msub>\n                      <mml:mi>A<\/mml:mi>\n                      <mml:mn>1<\/mml:mn>\n                    <\/mml:msub>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mo>\u22ef<\/mml:mo>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>A<\/mml:mi>\n                      <mml:mi>n<\/mml:mi>\n                    <\/mml:msub>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> conditioned on the side information <jats:italic>E<\/jats:italic> at the end of the process can be bounded from below by a sum of von Neumann entropies associated with the individual steps. This is a generalisation of the entropy accumulation theorem (EAT) (Dupuis et al. in Commun Math Phys 379: 867\u2013913, 2020), which deals with a more restrictive model of side information: there, past side information cannot be updated in subsequent rounds, and newly generated side information has to satisfy a Markov condition. Due to its more general model of side-information, our generalised EAT can be applied more easily and to a broader range of cryptographic protocols. As examples, we give the first multi-round security proof for blind randomness expansion and a simplified analysis of the E91 QKD protocol. 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