A novel meshless reconstruction algorithm for digital tomosynthesis (DT) is presented and assessed against experimental data. The algorithm does not require a three-dimensional grid or mesh allocation and performs a slice-by-slice reconstruction where each slice position can be chosen at runtime. The methodology is based on the filtered backprojection algorithm adapted to DT. However, in the traditional approach the backprojection comes first and the filtering follows. Because the backprojection requires ray tracing, in our case it is replaced with an equivalent image mapping procedure. The idea to swap the filtering and backprojection had been introduced earlier for computerized tomography (CT). Here we use this idea but develop it differently. Contrary to CT imaging, where the source and detector are rotated, in DT the subject and the flat panel detector are fixed in space. This imaging geometry allows reconstruction in planes parallel to the flat panel detector, which results in a significant simplification of the filter of backprojection algorithm. Moreover, the algorithm is not memory demanding and can be used with very large datasets. Two versions of the meshless algorithm are presented. One of them is based on convolution type filtering, while another uses filtering in the Fourier domain. Both versions are assessed and compared against the cone beam algorithm.