Kodaira embedding theorem says: suppose $M$ is a Kähler manifold with a positive line bundle $L$, then there exists a sufficiently large number $m$ such that basis of $H^0(M,L^{\otimes m})$ give embedding of $M$ into some projective space $\mathbb{CP}^N$, which we denote by $\phi_m: M \rightarrow \mathbb{CP}^N$.
Now I want to propose the following question: suppose $m_1,m_2$ are sufficiently large such that $L^{\otimes m_1}, L^{\otimes m_2}$ are both very ample, then what can we say about the relationship between two Kodaira embedding $\phi_{m_1}, \phi_{m_2}$?
Maybe above question is too general and seems less approachable. Let me change to ask the following question, which will be enough for my usage: suppose $m_2=d \cdot m_1$ where $d$ is an integer. Can we prove $\phi_{m_1}, \phi_{m_2}$ are connected via d-uple embedding?
Recall d-uple embedding is given by: let $N_2 = \binom{N_1+d}{d}-1$, then d-uple embedding is $\iota: \mathbb{CP}^{N_1}\rightarrow \mathbb{CP}^{N_2}, [z_0:z_1:\cdots:z_{N_1}] \mapsto [z_0^d: z_0^{d-1}z_1:\cdots:z_n^{d}]$, is $\phi_{m_2}= \iota \circ \phi_{m_1}$ correct?
Thank you in advance.
Edit: Thanks to Ted's advise in comment, I try to start from simple example to get some ideas. Let $M=\mathbb{CP}^n$ with hyperplane bundle $L=\mathcal{O}_{\mathbb{CP^n}}(m)$ where $m>0$.
Note global section $H^0(\mathbb{CP}^n, \mathcal{O}(m))$ is generated by homogeneous polynomials with degree $k$, then $\text{dim}_{\mathbb{C}} H^0(\mathbb{CP}^n, \mathcal{O}(m))=\binom{n+m}{n}$. In the above setting, Kodaira embedding is exactly given by k-uple embedding: $\phi: \mathbb{CP}^{n}\rightarrow \mathbb{CP}^{\binom{n+m}{n}-1}, [z_0:z_1:\cdots:z_{n}] \mapsto [z_0^m: z_0^{m-1}z_1:\cdots:z_n^{m}]$.
Now, if $m_2 = d \cdot m_1$, then $\phi_{m_2}= \iota \circ \phi_{m_1}$, where $\iota$ is d-uple embedding. And as Lazzaro pointed out in the comment, it is not true the corresponding maps are related by a d-uple embedding when $m_2 \neq d \cdot m_1$.