The naive view of (say, complex) cobordism of a space $X$ is that it should be the group generated by continuous families of closed manifolds parametrized by $X$. Acting even more naively, one may try to represent elements of this group by continuous maps $p:Y\to X$ together with compatible (in suitable sense) manifold structures on fibres $p^{-1}(x)$ for all $x\in X$. And if one's naivety borders on downright stupidity one will try to define all this in such way that for a continuous path $\gamma:[0,1]\to X$ the pullback $\gamma^*(p):[0,1]\times_XY\to[0,1]$ will be a bordism-equipped-with-a-Morse-function -- in particular, its total space will be a manifold with boundary equal to the disjoint union of $p^{-1}\gamma(0)$ and $p^{-1}\gamma(1)$.
The main obstacle to doing all this can be seen in that last point: along the way between 0 and 1, fibres will undergo (in best case) some surgeries and it is not clear how to incorporate these into the notion of continuous family of manifolds.
Probably the correct immediate reaction to this must be that there is a well known proper way to do it, contained in the classical paper "Elementary Proofs of Some Results of Cobordism Theory Using Steenrod operations" by Quillen. That is, presume that $p$ is a map of manifolds and equip $p$ with an embedding into a vector bundle $E\to X$ over $X$ together with appropriate structure on the normal bundle of $Y$ inside $E$.
My question then is,
Does anybody know about some sort of "tightening" Quillen's construction so as to allow $X$ to be more general space (say, finite CW-complex), and the result coming closer to the above naive picture?
What I tried to figure out was long ago suggested to me in a conversation by Michael Joachim - namely, to try using the concept of stratifold by Kreck: something like endowing $X$ with a cell structure such that only fibres over vertices would be manifolds, while fibres over points in cells of higher and higher dimensions would be allowed to have more and more severe singularities, whereas surgeries of fibres would occur when crossing cell boundaries, by rearranging strata in some way.
However I definitely lack any expertise to implement this, and then also there might be entirely different ways to do it.
Added slightly later on the afterthought - concerning my tries with cell structures. Probably more consistent approach is to "turn around" this picture and request that fibres which are manifolds are over points in cells of highest dimension, fibres with "easiest" singularities are allowed along cells of codimension one, etc.
In fact I don't even know whether anything forbids assuming that all fibres are manifolds and just the map deviates more and more from being fibration along cells of higher and higher codimension...
