Multiple, Lit Plume, Mute Pill and so on
Python 2, 48 bytes



lambda k,N:[k*(10**N**k+10**i)for i in range(N)]


Try it online!

For example, k=12,n=4 produces this list of numbers, written out in lines for clarity, with the ...'s hiding a big string of zeroes:



1200...0000012
1200...0000120
1200...0001200
1200...0012000


So, we have a prefix of k, followed by another k in a position that moves one place left each time, and the rest zeroes. We need the prefix because leading zeroes aren't allowed for the anagrams, so we make it so that all the outputs have the same number of digits.

We leave enough zeroes to ensure that the right string to never bump into the left one. N**k zeroes more than suffices for this, giving extremely big numbers; N+k or even N+log_10(k) would be enough. Note that Python right-associates 10**N**k as 10**(N**k).

Unfortunately this variable bound doesn't lend well to recursive solutions that update k as they go. An upper bound on how big k can be, even if enormous, would simplify this.