Let $N=\mathbb{S}^n$, $M \subseteq \mathbb{S}^n$ be a hemisphere, including the equator (I am considering $M$ with boundary).
Let $f_1:M \to N$ be the inclusion map, and let $f_2:M \to N$ be the reflection of $\mathbb{S}^n$ w.r.t $\partial M$.
How to show $f_1,f_2$ are not homotopic through a boundary respecting homotopy?
i.e I want to show there is no homotopy which is constant on the equator $\partial M$.
(Can we use degree here? According to wikipedia, the notion of a degree is defined only in case where the source is without boundary, or if the source has a boundary, then target must has one as well and the map must send boundary to boundary. In our case, the source has a boundary but the target does not).
Ah. OK. So you want to show, for instance, that the map $[0, 1] \to S^1 : x \mapsto (x, \sqrt{1-x^2})$ is not homotopic, rel boundary, to $[0, 1] \to S^1 : x \mapsto (x, -\sqrt{1-x^2})$
Such a map passes naturally to the quotient. $M$ is just the disk $D^n$, so you get $f_i' : D^n /\partial D^n \to S^n / \partial M$, i.e., $$ f_i' : S^n \to S^n / \partial M. $$ The space $K$ on the right is the union of two $n$-spheres that meet at a single point (i.e., a sphere with its belt snugged up). Now $H_n(K) = Z \oplus Z$, by the homology sequence of a pair, with $U$ being the (image under the quotient map of) upper sphere plus the arctic region of the lower one, and $V$ being the lower sphere plus the antarctic region of the upper. More formally: $$ U = \{ (x_0, \ldots x_n) | x_n > -\frac{1}{2} \}\\ V = \{ (x_0, \ldots x_n) | x_n < +\frac{1}{2} \}. $$
Furthermore, the induced maps on homology are $Z \to Z \oplus Z : u \mapsto (1, 0)$ and $u \mapsto (0, 1)$. And since homotopic maps induce the same map on homology, $f_1'$ and $f_2'$ are evidently not homotopic.