Selfmates with active white king 10

a) 1.Sh3! Qa3 2.Kg1 Qa4 3.Kf1 Qa3 4.Ke1 Qa4 5.Kd2 Qa3 6.Kc1 Qa4 7.Kxb2 Qa3#

b) 1.Kh2! Qa4 2.Kg3 Qa3 3.Kf4 Qa4 4.Kxe4 Qa3 5.Kd5 Qa4 6.Kc6 Qa3 7.Kxb5 Qa4#

We can see typical oscillation in one solution, however to have two different finales after oscillations of Black and marches of wK, it is unusual.

1.Kg1 Bh4 2.Kf1 Bg5 3.Ke2 4.Kd3 5.Kc4 6.Kc5 7.Sd6 8.Kc6 9.Kc7 10.Se8 11.Kd8 Bh4 12.Qc7+ Rd6#

Here the oscillation is made by bB and on his way to d8, wK is temporarily shielded by wS. A mate is given by doublecheck.

1.Be5! Kc5 2.Kf6 Kc6 3.Kf5 Kc5 4.Ke4 Kc6 5.Kd3 Kc5 6.Kc2 Kc6 7.Kb2 Kc5 8.Ka2 Kc6 9.Be4+ Kc5 10.Rb2 b3+ 11.Ka3 b4+ 12.Ka4 b5+ 13.Ka5 b6#

In this problem the mate is given by an indirect battery after wK's march of 11 moves, interrupted by wB freezing bK and thus forcing the series of bp moves.

1.Bd4! Kb8 2.Kf7 Ka8 3.Ke6 Kb8 4.Kd5 Ka8 5.Kc4 Kb8 6.Kb3 Ka8 7.Ka2 Kb8 8.Kb1 Ka8 9.Kc1 Kb8 10.Kd1 Ka8 11.Ke1 Kb8 12.Rd1 Ka8 13.Sc7+ Kb8 14.Sa6+ Ka8 15.Bf2 gxf2#

What I don't understand here is: why the author has chosen the key taking the bK flight...?

1.Ke7! Ka5 2.Kf6 Ka4 3.Kg5 Ka5 4.Kxh4 Ka4 5.Kg4 Ka5 6.h4 Ka4 7.Kh3 Ka5 8.Kh2 Ka4 9.Kg1 Ka5 10.Kf1 Ka4 11.Ke1 Ka5 12.Kd2 Ka4 13.Kd3 Ka5 14.Rd2 Ka4 15.Sc4 bxc4#

White king makes 12 moves here, but they are interrupted by wph3.

1.Kf6! Ke4 2.Qe5+ Kf3 3.Qg5 Ke4 4.Kg6 Kf3 5.Kh5 Ke4 6.Kh4 Kf3 7.Kh3 Ke4 8.Kh2 Kf3 9.Kg1 Ke4 10.Kf1 Kf3 11.Rf2+ Ke4 12.Qf5+ Ke3 13.Re2+ dxe2+ 14.Ke1 d3 15.Qg4 d2#

A miniature, in which White first creates a cage for bK, then sends his king to the first rank and finally releases mating pawns.

1.d8B! g6 2.Bdc7 h4 3.Bb8 h3 4.Bxa7 Raxa7 5.Kf1 Ra6 6.Ke2 Raa7 7.Kd3 Ra6 8.Kc4 Raa7 9.Kd5 Ra6 10.Ke6 Raa7 11.Kf6 Ra6 12.Kxg6 Raa7 13.Kh6 Ra6 14.g6 Raa7 15.g7 Ra6 16.Kh7 Raa7 17.Kh8 Ra6 18.g8B Raa7 19.Qc8+ Rb8 20.Bd5+ Rab7 21.Kg8 Rxc8#

Here White at the beginning gives Black an oscillation space, then runs with the king and after his arrival on the 8th rank the finale follows. Two bishop promotions on the way.