A celebrated result of Bertolini–Darmon–Prasanna shows that certain Rankin–Selberg $p$-adic $L$-functions — constructed via $p$-adic interpolation of the Waldspurger formula — can be evaluated at points outside their interpolation range (which we refer to as wall-crossing) in terms of Generalized Heegner cycles (and serve as $p$-adic analogues of first derivatives at the central critical point). This principle has been extended to triple products by Bertolini–Seveso–Venerucci and Darmon–Rotger, who relate values of Hsieh’s unbalanced $p$-adic $L$-functions (constructed $p$-adically interpolating Ichino and Hsieh’s explicit GGP formulae) on the balanced range to diagonal cycles. I will report on a result where wall-crossing is used to factor a triple product $p$-adic $L$-function in a setting with an empty interpolation range — yielding a $p$-adic Artin formalism for families of the form $f \times g\times g$. The key input is the arithmetic Gan–Gross–Prasad (Gross–Kudla) conjecture, linking central derivatives of (complex) triple product $L$-functions to Bloch–Beilinson heights of diagonal cycles and their comparison with their $GL(2)$-counterpart (Gross–Zagier formula). I will also discuss an extension to families on $GSp(4) \times GL(2) \times GL(2)$, where a new double wall-crossing phenomenon arises and is required to explain a $p$-adic Artin formalism for families of the form $F \times g \times g$. This suggests a higher BDP/arithmetic GGP formulae concerning second-order derivaties. This talk is based on four disjoint projects, joint with D. Casazza, A. Pal, O. Rivero, R. Sakamoto, and C. de Vera Picquero.