Accelerating　beams　have　been　the　subject　of　extensive　research　in　the　last　few　decades　because　of　their　self-acceleration　and　diffraction-free　propagation　over　several　Rayleigh　lengths.　Here,　we　investigate　the　propagation　dynamics　of　a　Fresnel　diffraction　beam　using　the　nonlocal　nonlinear　Schrödinger　equation　(NNLSE).　When　a　nonlocal　nonlinearity　is　introduced　into　the　linear　Schrödinger　equation　without　invoking　an　external　potential,　the　evolution　behaviors　of　incident　Fresnel　diffraction　beams　are　modulated　regularly,　and　certain　novel　phenomena　are　observed.　We　show　through　numerical　calculations,　under　varying　degrees　of　nonlocality,　that　nonlocality　significantly　affects　the　evolution　of　Fresnel　diffraction　beams.　Further,　we　briefly　discuss　the　two-dimensional　case　as　the　equivalent　of　the　product　of　two　one-dimensional　cases.　At　a　critical　point,　the　Airy-like　intensity　profile　oscillates　between　the　first　and　third　quadrants,　and　the　process　repeats　during　propagation　to　yield　an　unusual　oscillation.　Our　results　are　expected　to　contribute　to　the　understanding　of　NNLSE　and　nonlinear　optics.　©　2021　Chinese　Physical　Society　and　IOP　Publishing　Ltd.