According to the Langlands functoriality conjecture, broadened to the setting of spherical varieties (of which reductive groups are special cases), a map between L -groups of spherical varieties should give rise to a functorial transfer of their local and automorphic spectra. The “beyond endoscopy” proposal predicts that this transfer will be realized as a comparison between limiting forms of the (relative) trace formulas of these spaces. In this paper, we establish the local transfer for the identity map between L -groups, for spherical affine homogeneous spaces X = H \ G whose dual group is SL 2 or PGL 2 (with G and H split). More precisely, we construct a transfer operator between orbital integrals for the ( X × X ) / G -relative trace formula, and orbital integrals for the Kuznetsov formula of PGL 2 or SL 2 . Besides the L -group, another invariant attached to X is a certain L -value, and the space of test measures for the Kuznetsov formula is enlarged to accommodate the given L -value. The transfer operator is given explicitly in terms of Fourier convolutions, making it suitable for a global comparison of trace formulas by the Poisson summation formula, hence for a uniform proof, in rank 1 , of the relations between periods of automorphic forms and special values of L -functions.