Let $M, N$ be Riemannian manifolds and $f: M \to N$ be a smooth map
(I'm actually only considering diffeomorphisms (flows)
$\Phi^t: M \to M$, but just for the sake of generality).

The first derivative of $f$ can be understood as its tangent map
$T f: T M \to T N$. Higher derivatives can abstractly be viewed as
maps between higher order tangent bundles.

I want to make estimates on the (operator norm) size of these higher
derivatives. In the higher order tangent spaces (see also the recent
question https://mathoverflow.net/questions/2019/) I'd have to use
induced metrics, which I don't readily know how to work with, and besides,
I think these would include the base, lower order derivatives as well.

I would prefer to keep things defined on the tangent/tensor bundle, in
a similar way as taking covariant derivatives for vector fields, but I
don't know how to do this for for maps $f: M \to N$.

So my question roughly is: are there natural/practical representations
of norms of higher order derivatives of maps between manifolds?

One thing I did come up with is representing $f$ in normal coordinates, as these are the most canonical charts and then use the norms in the tangent spaces at the argument and image points $x$ and $y = f(x)$.

(The basis for this question is that I want to obtain a Gronwall-like
growth estimate for the higher derivatives of a flow $\Phi^t$ in
terms of the exponential growth of its tangent flow $D \Phi^t$.)