We consider pairs of Lagrangian submanifolds $(L_0,L), (L_1, L)$ belonging to
the class of Lagrangian submanifolds with \emph{conic} ends on \emph{Weinstein
manifolds}. The main purpose of the present paper is to define a canonical
chain map $h_\CL: CF(L_0,L) \to CF(L_1,L)$ of Lagrangian Floer complex inducing
an isomorphism in homology, under the Hamiltonian isotopy $\CL=\{L_s\}_{0 \leq
s\leq 1}$ generated by \emph{conic} Hamiltonian functions such that the
intersections $L \cap L_s$ do not escape to infinity. The main ingredients of
the proof is an a priori bound for general isotopy of the energy
\emph{quadratic} at infinity and a $C^0$-bound for the \emph{$C^1$-small}
isotopy $\CL = \{L_s\}$, for the associated pseudo-holomorphic map equations
with \emph{moving} Lagrangian boundary induced by a conic Hamiltonian isotopy.
For the Lagrangian submanifolds with \emph{asymptotically conic} ends, we
construct a natural homomorphism $h_\LL: HF(L_0,L) \to HF(L_1,L)$ for which the
corresponding chain map may \emph{not} necessarily exist.

This provides a more conventional construction of the chain isomorphism which
replaces the sophisticated method using the Lagrangian cobordism via the
machinery of \cite{kasturi-oh1,kasturi-oh2} whose details were only outlined in
\cite{oh:gokova}.